A317912 Expansion of Product_{k>=2} 1/(1 - k*x^k).

1, 0, 2, 3, 8, 11, 31, 41, 101, 156, 318, 498, 1037, 1555, 3024, 4889, 8849, 14112, 25622, 40322, 71314, 113926, 194677, 310819, 530030, 835484, 1400523, 2226307, 3668998, 5797558, 9521310, 14942262, 24298136, 38187102, 61384028, 96161997, 154078991, 239891926, 381723396

Offset

0,3

Comments

First differences of A006906.

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Index entries for sequences related to partitions

Formula

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

Example

a(6) = 31 because we have [6], [4, 2], [3, 3], [2, 2, 2] and 6 + 4*2 + 3*3 + 2*2*2 = 31.

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)))
    end:
a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 10 2018

Mathematica

nmax = 38; CoefficientList[Series[Product[1/(1 - k x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[Exp[Sum[Sum[k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 38}]]

Crossrefs

Cf. A002865, A006906, A191659, A302830.

Sequence in context: A143914 A041123 A042045 * A096506 A119155 A119195

Adjacent sequences: A317909 A317910 A317911 * A317913 A317914 A317915

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 10 2018

Status

approved