%I #77 Apr 29 2023 13:59:17

%S 0,1,1,2,2,4,5,8,10,16,20,29,37,52,66,90,113,151,190,248,310,400,497,

%T 632,782,985,1212,1512,1851,2291,2793,3431,4163,5084,6142,7456,8972,

%U 10836,12989,15613,18646,22316,26561,31659,37556,44601,52743,62416,73593,86809,102064,120025,140736

%N Number of partitions of n into an odd number of parts.

%C Number of partitions of n in which greatest part is odd.

%C Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].

%C Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - _Emeric Deutsch_, Apr 05 2006

%C Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities. Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - _Clark Kimberling_, Feb 01 2014, corrected Jan 06 2021

%C In Chaves 2011 see page 38 equation (3.20). - _Michael Somos_, Dec 28 2014

%C 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 A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - _Clark Kimberling_, Dec 15 2016

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.

%H T. D. Noe, <a href="/A027193/b027193.txt">Table of n, a(n) for n = 0..999</a>

%H Roland Bacher and P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685/document">Conjugacy growth series of some infinitely generated groups</a>, hal-01285685v2, 2016.

%H D. R. C. Chaves, <a href="http://hdl.handle.net/10183/29250">Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan</a>, 2011. In Portuguese.

%H 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_0(n).

%F a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.

%F For g.f. see under A027187.

%F G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - _Emeric Deutsch_, Apr 05 2006

%F G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - _Joerg Arndt_, Feb 02 2014

%F G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - _Michael Somos_, Dec 28 2014

%F a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - _Reinhard Zumkeller_, Apr 22 2006

%e G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...

%e From _Gus Wiseman_, Feb 11 2021: (Start)

%e The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (111) (211) (221) (222) (322) (332)

%e (311) (321) (331) (422)

%e (11111) (411) (421) (431)

%e (21111) (511) (521)

%e (22111) (611)

%e (31111) (22211)

%e (1111111) (32111)

%e (41111)

%e (2111111)

%e The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.

%e (1) (11) (3) (31) (5) (33) (7) (53)

%e (111) (1111) (32) (51) (52) (71)

%e (311) (321) (322) (332)

%e (11111) (3111) (331) (521)

%e (111111) (511) (3221)

%e (3211) (3311)

%e (31111) (5111)

%e (1111111) (32111)

%e (311111)

%e (11111111)

%e (End)

%p g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # _Emeric Deutsch_, Apr 05 2006

%t nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Dec 01 2012 *)

%t a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* _Michael Somos_, Dec 28 2014 *)

%t a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* _Michael Somos_, Dec 28 2014 *)

%t a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* _Michael Somos_, Dec 28 2014 *)

%t a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* _Michael Somos_, Dec 28 2014 *)

%o (PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* _Michael Somos_, Jul 24 2012 */

%o (PARI) q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ _Joerg Arndt_, Mar 23 2014

%Y Cf. A000041, A000700, A000701, A026837, A046682.

%Y The Heinz numbers of these partitions are A026424 or A244991.

%Y The even-length version is A027187.

%Y The case of odd sum as well as length is A160786, ranked by A340931.

%Y The case of odd maximum as well as length is A340385.

%Y Other cases of odd length:

%Y - A024429 counts set partitions of odd length.

%Y - A067659 counts strict partitions of odd length.

%Y - A089677 counts ordered set partitions of odd length.

%Y - A166444 counts compositions of odd length.

%Y - A174726 counts ordered factorizations of odd length.

%Y - A332304 counts strict compositions of odd length.

%Y - A339890 counts factorizations of odd length.

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A026804 counts partitions whose least part is odd.

%Y A058695 counts partitions of odd numbers, ranked by A300063.

%Y A072233 counts partitions by sum and length.

%Y A101707 counts partitions of odd positive rank.

%Y Cf. A078408, A174725, A236914, A300272, A340831.

%K nonn

%O 0,4

%A _Clark Kimberling_