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A027187 - OEIS
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[Edited by <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Jan 05 2021]</div> <div class=sectline>Number of partitions of n+1 into an odd number of parts, the least being 1.</div> <div class=sectline>Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at <a href="/A027193" title="Number of partitions of n into an odd number of parts.">A027193</a>. - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Feb 01 2014, corrected Jan 06 2021</div> <div class=sectline>Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has <a href="/A027193" title="Number of partitions of n into an odd number of parts.">A027193</a>(n) negative coefficients, <a href="/A027187" title="Number of partitions of n into an even number of parts.">A027187</a>(n) positive coefficients, and <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a> terms. The maximal coefficient in d(n), in absolute value, is <a href="/A102462" title="Max{ k!/(a(1)!*a(2)!*..*a(n)!) : a(1) + 2*a(2) + 3*a(3) + ... + n*a(n) = n, a(1) + a(2) + ... + a(n) = k }.">A102462</a>(n). - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Dec 15 2016</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Alois P. Heinz, <a href="/A027187/b027187.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)</div> <div class=sectline>George E. Andrews and David Newman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.html">The Minimal Excludant in Integer Partitions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.</div> <div class=sectline>Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, <a href="https://arxiv.org/abs/2406.06036">How large is the character degree sum compared to the character table sum for a finite group?</a>, arXiv:2406.06036 [math.RT], 2024. See p. 13.</div> <div class=sectline>Roland Bacher and Pierre De La Harpe, <a href="https://hal.science/hal-01285685">Conjugacy growth series of some infinitely generated groups</a>, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)</div> <div class=sectline>N. J. Fine, <a href="http://www.jstor.org/stable/2307653">Problem 4314</a>, Amer. Math. Monthly, Vol. 57, 1950, 421-423.</div> <div class=sectline>Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).</div> <div class=sectline>Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (<a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>(n) + (-1)^n * <a href="/A000700" title="Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts; number of self-conjugate part...">A000700</a>(n))/2.</div> <div class=sectline>a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>. [Fine] - <a href="/wiki/User:David_Callan">David Callan</a>, Mar 14 2004</div> <div class=sectline>From <a href="/wiki/User:Bill_Gosper">Bill Gosper</a>, Jun 25 2005: (Start)</div> <div class=sectline>G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3 q^4 + 3 q^5 + 6 q^6 + ...</div> <div class=sectline>= Sum_{n >= 0} q^(2n)/(q; q)_{2n}</div> <div class=sectline>= ((Product_{k >= 1} 1/(1-q^k) + (Product_{k >= 1} 1/(1+q^k))/2.</div> <div class=sectline>Also, let B(q) = Sum_{n >= 0} <a href="/A027193" title="Number of partitions of n into an odd number of parts.">A027193</a>(n) q^n = q + q^2 + 2 q^3 + 2 q^4 + 4 q^5 + 5 q^6 + ...</div> <div class=sectline>Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Product_{k >= 1} 1/(1-q^k) - (Product_{k >= 1} 1/(1+q^k))/2.</div> <div class=sectline>Also we have the following identity involving 2 X 2 matrices:</div> <div class=sectline>Product_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]</div> <div class=sectline>= [ A(q) B(q) / B(q) A(q) ]. (End)</div> <div class=sectline>a(2*n) = <a href="/A046682" title="Number of cycle types of conjugacy classes of all even permutations of n elements.">A046682</a>(2*n), a(2*n+1) = <a href="/A000701" title="One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.">A000701</a>(2*n+1); a(n) = <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a>(n)-<a href="/A027193" title="Number of partitions of n into an odd number of parts.">A027193</a>(n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 22 2006</div> <div class=sectline>Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2006</div> <div class=sectline>G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2006</div> <div class=sectline>a(n) = <a href="/A338914" title="Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.">A338914</a>(n) + <a href="/A096373" title="Number of partitions of n such that the least part occurs exactly twice.">A096373</a>(n). - <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Jan 06 2021</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...</div> <div class=sectline>From <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Jan 05 2021: (Start)</div> <div class=sectline>The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by <a href="/A028260" title="Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda...">A028260</a>.</div> <div class=sectline> (11) (21) (22) (32) (33) (43) (44)</div> <div class=sectline> (31) (41) (42) (52) (53)</div> <div class=sectline> (1111) (2111) (51) (61) (62)</div> <div class=sectline> (2211) (2221) (71)</div> <div class=sectline> (3111) (3211) (2222)</div> <div class=sectline> (111111) (4111) (3221)</div> <div class=sectline> (211111) (3311)</div> <div class=sectline> (4211)</div> <div class=sectline> (5111)</div> <div class=sectline> (221111)</div> <div class=sectline> (311111)</div> <div class=sectline> (11111111)</div> <div class=sectline>The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by <a href="/A244990" title="After 1, numbers whose greatest prime factor is a prime with an even index; n such that A061395(n) is even.">A244990</a>.</div> <div class=sectline> (2) (21) (4) (41) (6) (43) (8)</div> <div class=sectline> (22) (221) (42) (61) (44)</div> <div class=sectline> (211) (2111) (222) (421) (62)</div> <div class=sectline> (411) (2221) (422)</div> <div class=sectline> (2211) (4111) (431)</div> <div class=sectline> (21111) (22111) (611)</div> <div class=sectline> (211111) (2222)</div> <div class=sectline> (4211)</div> <div class=sectline> (22211)</div> <div class=sectline> (41111)</div> <div class=sectline> (221111)</div> <div class=sectline> (2111111)</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Mar 13 2012 *)</div> <div class=sectline>a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, May 06 2015 *)</div> <div class=sectline>a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, May 06 2015 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2006 */</div> <div class=sectline>(PARI) q='q+O('q^66); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Mar 23 2014</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000701" title="One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.">A000701</a>, <a href="/A046682" title="Number of cycle types of conjugacy classes of all even permutations of n elements.">A046682</a>, <a href="/A026838" title="Number of partitions of n into distinct parts, the greatest being even.">A026838</a>, <a href="/A102462" title="Max{ k!/(a(1)!*a(2)!*..*a(n)!) : a(1) + 2*a(2) + 3*a(3) + ... + n*a(n) = n, a(1) + a(2) + ... + a(n) = k }.">A102462</a>.</div> <div class=sectline>The Heinz numbers of these partitions are <a href="/A028260" title="Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda...">A028260</a>.</div> <div class=sectline>The odd version is <a href="/A027193" title="Number of partitions of n into an odd number of parts.">A027193</a>.</div> <div class=sectline>The strict case is <a href="/A067661" title="Number of partitions of n into distinct parts such that number of parts is even.">A067661</a>.</div> <div class=sectline>The case of even sum as well as length is <a href="/A236913" title="Number of partitions of 2n of type EE (see Comments).">A236913</a> (the even bisection).</div> <div class=sectline>Other cases of even length:</div> <div class=sectline>- <a href="/A024430" title="Expansion of e.g.f. cosh(exp(x)-1).">A024430</a> counts set partitions of even length.</div> <div class=sectline>- <a href="/A034008" title="a(n) = floor(2^|n-1|/2). Or: 1, 0, followed by powers of 2.">A034008</a> counts compositions of even length.</div> <div class=sectline>- <a href="/A052841" title="Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).">A052841</a> counts ordered set partitions of even length.</div> <div class=sectline>- <a href="/A174725" title="a(n) = (A074206(n) + A008683(n))/2.">A174725</a> counts ordered factorizations of even length.</div> <div class=sectline>- <a href="/A332305" title="Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.">A332305</a> counts strict compositions of even length</div> <div class=sectline>- <a href="/A339846" title="Number of even-length factorizations of n into factors > 1.">A339846</a> counts factorizations of even length.</div> <div class=sectline><a href="/A000009" title="Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd p...">A000009</a> counts partitions into odd parts, ranked by <a href="/A066208" title="All primes that divide n are of the form prime(2k-1), where prime(k) is k-th prime.">A066208</a>.</div> <div class=sectline><a href="/A026805" title="Number of partitions of n in which the least part is even.">A026805</a> counts partitions whose least part is even.</div> <div class=sectline><a href="/A072233" title="Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguish...">A072233</a> counts partitions by sum and length.</div> <div class=sectline><a href="/A101708" title="Number of partitions of n having positive even rank (the rank of a partition is the largest part minus the number of parts).">A101708</a> counts partitions of even positive rank.</div> <div class=sectline>Cf. <a href="/A000700" title="Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts; number of self-conjugate part...">A000700</a>, <a href="/A026424" title="Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.">A026424</a>, <a href="/A058696" title="Number of ways to partition 2n into positive integers.">A058696</a>, <a href="/A096373" title="Number of partitions of n such that the least part occurs exactly twice.">A096373</a>, <a href="/A244990" title="After 1, numbers whose greatest prime factor is a prime with an even index; n such that A061395(n) is even.">A244990</a>, <a href="/A300061" title="Heinz numbers of integer partitions of even numbers.">A300061</a>.</div> <div class=sectline>Sequence in context: <a href="/A325834" title="Number of integer partitions of n whose number of submultisets is less than or equal to n.">A325834</a> <a href="/A365924" title="Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.">A365924</a> <a href="/A241832" title="Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) > number o...">A241832</a> * <a href="/A056508" title="Number of periodic palindromic structures of length n using exactly two different symbols.">A056508</a> <a href="/A050065" title="a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), sta...">A050065</a> <a href="/A367394" title="Number of integer partitions of n whose length is a semi-sum of the parts.">A367394</a></div> <div class=sectline>Adjacent sequences: <a href="/A027184" title="a(n) = (1/2)*(n-th largest even number in array T given by A027170).">A027184</a> <a href="/A027185" title="Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the least being k.">A027185</a> <a href="/A027186" title="Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the least being k.">A027186</a> * <a href="/A027188" title="a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n int...">A027188</a> <a href="/A027189" title="Number of partitions of n into an odd number of parts, the least being 3; also, a(n+3) = number of partitions of n into an e...">A027189</a> <a href="/A027190" title="Number of partitions of n into an odd number of parts, the least being 4; also, a(n+4) = number of partitions of n into an e...">A027190</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Offset changed to 0 by <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jul 24 2012</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified December 12 09:24 EST 2024. 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