A304620 - OEIS

%I #27 Jun 27 2021 07:52:15

%S 1,1,2,3,6,9,15,22,34,48,70,97,137,186,255,341,459,605,800,1042,1359,

%T 1751,2256,2879,3672,4645,5869,7367,9234,11508,14319,17730,21916,

%U 26975,33143,40570,49575,60376,73402,88974,107666,129933,156546,188148,225767,270300,323115,385453

%N Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).

%C Partial sums of A027187.

%C From _Gus Wiseman_, Jun 26 2021: (Start)

%C Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:

%C   1   111   32      331       54          551           76

%C             11111   3211      3222        3332          5422

%C                     1111111   3321        5411          5521

%C                               33111       33221         33331

%C                               321111      322211        55111

%C                               111111111   332111        322222

%C                                           3311111       332221

%C                                           32111111      333211

%C                                           11111111111   541111

%C                                                         3322111

%C                                                         32221111

%C                                                         33211111

%C                                                         331111111

%C                                                         3211111111

%C                                                         1111111111111

%C Also odd-length partitions of 2n+1 with exactly one odd part.

%C (End)

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = A000070(n) - A306145(n).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - _Vaclav Kotesovec_, Aug 20 2018

%t nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]

%t nmax = 47; CoefficientList[Series[(1 + EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]

%t Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman_, Jun 26 2021 *)

%Y First differences are A027187.

%Y The version for even instead of odd greatest part is A306145.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A000070 counts partitions with alternating sum 1.

%Y A067661 counts strict partitions of even length.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, A338907, A344611.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 19 2018