A317913 Expansion of Product_{k>=2} (1 + k*x^k).

1, 0, 2, 3, 4, 11, 14, 29, 35, 85, 101, 187, 276, 419, 686, 1042, 1483, 2258, 3517, 4727, 7720, 10582, 15842, 21985, 32744, 45586, 65598, 93940, 131684, 183731, 260977, 357689, 500476, 699225, 946851, 1342110, 1808841, 2495154, 3375385, 4657186, 6224608, 8524443, 11468183, 15428030

Offset

0,3

Comments

Sum of products of terms in all partitions of n into distinct parts >= 2.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Index entries for sequences related to partitions

Formula

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

a(n) + a(n-1) = A022629(n). - Vaclav Kotesovec, Aug 21 2018

Example

a(7) = 29 because we have [7], [5, 2], [4, 3] and 7 + 5*2 + 4*3 = 29.

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+ i*b(n-i, min(n-i, i-1)))
    end:
a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 10 2018

Mathematica

nmax = 43; CoefficientList[Series[Product[(1 + k x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[Exp[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

Crossrefs

Cf. A022629, A025147, A298598, A317912.

Sequence in context: A176541 A295721 A171376 * A141704 A061919 A328883

Adjacent sequences: A317910 A317911 A317912 * A317914 A317915 A317916

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 10 2018

Status

approved