A302830 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - k*x^k).

1, 2, 5, 11, 25, 50, 106, 203, 401, 755, 1427, 2597, 4804, 8566, 15352, 27027, 47551, 82187, 142445, 243025, 414919, 700739, 1181236, 1972552, 3293898, 5450728, 9008081, 14791741, 24244399, 39494615, 64266141, 103979929, 167991853, 270190879, 433773933, 693518984

Offset: 0

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

• nonn

Comments

Partial sums of A006906.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A000070, A006906, A091360, A302831.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

Mathematica

nmax = 35; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

Formula

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

From Vaclav Kotesovec, Apr 14 2018: (Start)

a(n) ~ c * 3^(n/3), where

c = 319343.48587983201292657132469068725642363369445... if mod(n,3)=0

c = 319343.34569378454521307030620964478962032866022... if mod(n,3)=1

c = 319343.21458897980925594955657564398036486423380... if mod(n,3)=2

(End)