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A260845 a(n) = Sum_{k=0..n} (-1)^k*P(n,k)*k!, where P(n,k) is the number of partitions of n into k parts.

%I A260845 #13 Sep 18 2019 14:28:50
%S A260845 1,-1,1,-5,21,-105,635,-4507,36457,-330971,3334377,-36913947,
%T A260845 445426739,-5818545721,81805507069,-1231690773053,19772941871385,
%U A260845 -337146625794753,6085005877228943,-115897323408009187,2323090928155541677,-48883768421712917555,1077440388662366900397
%N A260845 a(n) = Sum_{k=0..n} (-1)^k*P(n,k)*k!, where P(n,k) is the number of partitions of n into k parts.
%H A260845 Alois P. Heinz, <a href="/A260845/b260845.txt">Table of n, a(n) for n = 0..449</a>
%F A260845 G.f.: Sum(n!*(-x)^n/Product(1-x^k, k=1..n), n=1..infinity).
%p A260845 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1,
%p A260845       b(n, i-1), 0)+expand(b(n-i, min(n-i, i))*x))
%p A260845     end:
%p A260845 a:= n-> (p-> add(i!*coeff(p, x, i)*(-1)^i, i=0..n))(b(n$2)):
%p A260845 seq(a(n), n=0..27);  # _Alois P. Heinz_, Sep 18 2019
%t A260845 CoefficientList[ Series[ Sum[ n!(-x)^n / Product[1 - x^k, {k, n}], {n, 0, 22}], {x, 0, 22}], x]
%o A260845 (Sage)
%o A260845 from sage.combinat.partition import number_of_partitions_length
%o A260845 [sum([(-1)^k*number_of_partitions_length(n,k)*factorial(k) for k in (0..n)]) for n in (0..22)]
%Y A260845 Cf. A008284, A101880.
%Y A260845 Row sums of A318144.
%K A260845 sign
%O A260845 0,4
%A A260845 _Peter Luschny_, Aug 01 2015