A302832 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^k)^k.

1, 2, 4, 9, 17, 33, 61, 110, 193, 335, 570, 955, 1582, 2586, 4185, 6706, 10646, 16757, 26178, 40587, 62503, 95637, 145445, 219929, 330766, 494898, 736858, 1092027, 1611185, 2367079, 3463490, 5048009, 7329935, 10605211, 15290942, 21973641, 31475620, 44946859, 63991639, 90842560

Offset: 0

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

• nonn

Comments

Partial sums of A026007.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to partitions

Crossrefs

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

Programs

Maple

b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n))
    end:
g:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+g(n)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

Mathematica

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

Formula

G.f.: (1/(1 - x))*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)).

From Vaclav Kotesovec, Apr 13 2018: (Start)

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)).

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

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

(End)