%I #9 Apr 14 2018 15:22:40
%S 1,2,4,9,16,31,56,99,163,283,469,757,1220,1915,3020,4748,7273,11014,
%T 16789,25033,37480,55782,82206,120033,174762,253092,364276,523814,
%U 749438,1064853,1509561,2128227,2986392,4186093,5832169,8121130,11272081,15576076,21446615,29479186,40360980
%N Expansion of (1/(1 - x))*Product_{k>=1} (1 + k*x^k).
%C Partial sums of A022629.
%H Alois P. Heinz, <a href="/A302831/b302831.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j).
%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, i*b(n-i, i-1))))
%p end:
%p a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Apr 13 2018
%t nmax = 40; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 40; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%Y Cf. A000009, A022629, A036469, A302830, A302832.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 13 2018
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