cp's OEIS Frontend

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.

%I A331517 #11 Jan 28 2020 05:48:21
%S A331517 1,0,1,3,13,59,336,2245,17408,153124,1505420,16342711,194060616,
%T A331517 2501178199,34766184181,518332353130,8250146291076,139618375340912,
%U A331517 2503167665128431,47393482639721484,944910760664087791,19787603213440946946,434229133448518143203
%N 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.
%H A331517 Vaclav Kotesovec, <a href="/A331517/b331517.txt">Table of n, a(n) for n = 0..400</a>
%F A331517 G.f.: Sum_{k>=0} !k * x^k / Product_{j=1..k} (1 - x^j).
%F A331517 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
%t A331517 Table[Sum[Length[IntegerPartitions[n, {k}]] Subfactorial[k], {k, 0, n}], {n, 0, 22}]
%t A331517 nmax = 22; CoefficientList[Series[Sum[Subfactorial[k] x^k/Product[(1 - x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%Y A331517 Cf. A000166, A008284, A101880, A331518.
%K A331517 nonn
%O A331517 0,4
%A A331517 _Ilya Gutkovskiy_, Jan 19 2020