A302832 - OEIS

%I #10 Apr 14 2018 14:49:14

%S 1,2,4,9,17,33,61,110,193,335,570,955,1582,2586,4185,6706,10646,16757,

%T 26178,40587,62503,95637,145445,219929,330766,494898,736858,1092027,

%U 1611185,2367079,3463490,5048009,7329935,10605211,15290942,21973641,31475620,44946859,63991639,90842560

%N Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^k)^k.

%C Partial sums of A026007.

%H Alois P. Heinz, <a href="/A302832/b302832.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_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)).

%F From _Vaclav Kotesovec_, Apr 13 2018: (Start)

%F a(n) ~ exp((3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/12) * 3^(2/3) * sqrt(Pi) * Zeta(3)^(1/6) * n^(1/3)).

%F a(n) ~ (2*n/(3*Zeta(3)))^(1/3) * A026007(n).

%F a(n) ~ erfi((3/2)^(2/3) * Zeta(3)^(1/6) * n^(1/3)) / 2^(13/12).

%F (End)

%p b:= proc(n) option remember;

%p       add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n))

%p     end:

%p g:= proc(n) option remember;

%p       `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)

%p     end:

%p a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+g(n)) end:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 13 2018

%t nmax = 39; CoefficientList[Series[1/(1 - x) Product[(1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 39; CoefficientList[1/(1 - x) Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

%Y Cf. A000009, A026007, A036469, A091360, A302831.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 13 2018