A302831 Expansion of (1/(1 - x))*Product_{k>=1} (1 + k*x^k).

1, 2, 4, 9, 16, 31, 56, 99, 163, 283, 469, 757, 1220, 1915, 3020, 4748, 7273, 11014, 16789, 25033, 37480, 55782, 82206, 120033, 174762, 253092, 364276, 523814, 749438, 1064853, 1509561, 2128227, 2986392, 4186093, 5832169, 8121130, 11272081, 15576076, 21446615, 29479186, 40360980

Offset: 0

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

• nonn

Comments

Partial sums of A022629.

Links

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

Crossrefs

Cf. A000009, A022629, A036469, A302830, A302832.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    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 = 40; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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]

Formula

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