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%I #14 Jul 20 2021 03:29:42
%S 0,1,2,3,4,5,7,9,12,16,21,27,35,44,55,69,85,104,127,154,186,224,268,
%T 320,381,452,534,630,741,869,1017,1187,1382,1606,1862,2155,2489,2869,
%U 3301,3792,4349,4979,5692,6497,7405,8429,9581,10876,12331,13963,15792,17840,20131,22691
%N Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).
%C Partial sums of A067659.
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.2 "Partitions into distinct parts", page 350.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = A036469(n) - A318155(n).
%F a(n) = A318155(n) - A078616(n).
%F a(n) ~ exp(Pi*sqrt(n/3)) * 3^(1/4) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, Aug 20 2018
%e From _Gus Wiseman_, Jul 18 2021: (Start)
%e Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
%e (2,1) (3,2) (4,3) (5,4) (6,5) (7,6) (8,7) (9,8)
%e (4,1) (5,2) (6,3) (7,4) (8,5) (9,6) (10,7)
%e (6,1) (7,2) (8,3) (9,4) (10,5) (11,6)
%e (8,1) (9,2) (10,3) (11,4) (12,5)
%e (10,1) (11,2) (12,3) (13,4)
%e (12,1) (13,2) (14,3)
%e (6,4,2,1) (14,1) (15,2)
%e (6,4,3,2) (16,1)
%e (8,4,2,1) (6,5,4,2)
%e (8,4,3,2)
%e (8,6,2,1)
%e (10,4,2,1)
%e Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
%e (End)
%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
%p end:
%p a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
%p seq(a(n), n=0..60);
%t nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
%t Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman_, Jul 18 2021 *)
%Y Partial sums of A067659.
%Y The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
%Y - Allowing any length gives A036469.
%Y - The non-strict version is A306145.
%Y - The version for odd length is A318155 (non-strict: A304620).
%Y - Allowing any number of odd parts gives A343942 (odd bisection of A067661).
%Y A000041 counts partitions.
%Y A027187 counts partitions of even length (strict: A067661).
%Y A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y Cf. A000070, A030229, A035294, A058696, A078616, A087447, A152146, A236559, A343941, A344611, A344739.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 19 2018
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