A331517 a(n) = Sum_{k=0..n} p(n,k) * !k, where p(n,k) = number of partitions of n into k parts and !k = subfactorial of k.

1, 0, 1, 3, 13, 59, 336, 2245, 17408, 153124, 1505420, 16342711, 194060616, 2501178199, 34766184181, 518332353130, 8250146291076, 139618375340912, 2503167665128431, 47393482639721484, 944910760664087791, 19787603213440946946, 434229133448518143203

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

G.f.: Sum_{k>=0} !k * x^k / Product_{j=1..k} (1 - x^j).

a(n) ~ exp(-1) * n! * (1 + 1/n + 2/n^2 + 5/n^3 + 16/n^4 + 60/n^5 + 253/n^6 + 1180/n^7 + 6023/n^8 + 33306/n^9 + 197719/n^10 + ...), for coefficients see A331826. - Vaclav Kotesovec, Jan 28 2020

Mathematica

Table[Sum[Length[IntegerPartitions[n, {k}]] Subfactorial[k], {k, 0, n}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[Sum[Subfactorial[k] x^k/Product[(1 - x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000166, A008284, A101880, A331518.

Sequence in context: A367057 A317776 A145942 * A340411 A074437 A026578

Adjacent sequences: A331514 A331515 A331516 * A331518 A331519 A331520

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 19 2020

Status

approved