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A227094 Binomial transform of A013999.

%I A227094 #16 Nov 02 2023 10:00:50
%S A227094 1,2,5,18,91,574,4199,34650,318645,3237034,36041657,436713506,
%T A227094 5722676895,80654047942,1216703923147,19562850695690,333991034593833,
%U A227094 6034449711055890,115036771019660269,2307582082535387570,48588759062255598563,1071533741191907032590
%N A227094 Binomial transform of A013999.
%F A227094 a(n) = sum(C(n,k)*A013999(k), k=0..n).
%F A227094 G.f.: sum(k!*x^(k-1)*(1-2*x)^k/(1-x)^(2*k), k>=1).
%F A227094 a(n) ~ n * n!. - _Vaclav Kotesovec_, Nov 02 2023
%p A227094 a:= proc(n) option remember; `if`(n<4, [1, 2, 5, 18][n+1],
%p A227094       (n+6)*a(n-1)-(5*n+7)*a(n-2)+(8*n-7)*a(n-3)-(4*n-12)*a(n-4))
%p A227094     end:
%p A227094 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 01 2013
%t A227094 a[n_] := a[n] = If[n < 4, {1, 2, 5, 18}[[n + 1]], (n + 6)*a[n - 1] - (5*n + 7)*a[n - 2] + (8*n - 7)*a[n - 3] - (4*n - 12)*a[n - 4]];
%t A227094 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 04 2018, from Maple *)
%o A227094 (Maxima) f(n):=sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!, k, 0, floor((n+1)/2)); a(n):=sum(binomial(n,k)*f(k), k, 0, n); makelist(a(n), n,0,20);
%Y A227094 Cf. A013999.
%K A227094 nonn
%O A227094 0,2
%A A227094 _Emanuele Munarini_, Jul 01 2013