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%I #44 Dec 11 2021 03:35:18
%S 1,2,4,13,54,278,1704,12126,98280,893904,9017280,99918120,1206500400,
%T 15768729360,221792780160,3340515069360,53641756586880,
%U 914849722725120,16514863528665600,314599179867396480,6306817346711481600,132727279189258656000
%N Expansion of Sum_{k>=0} k! * x^k * (1 + x)^(k+1).
%H Seiichi Manyama, <a href="/A346550/b346550.txt">Table of n, a(n) for n = 0..449</a>
%F a(n) = Sum_{k=floor(n/2)..n} k! * binomial(k+1,n-k).
%F a(n) = A240172(n-1) + A240172(n) for n > 0.
%F a(n) = (n-2) * a(n-1) + 2 * (n-1) * a(n-2) + (n-2) * a(n-3) for n > 2.
%F a(n) ~ exp(1) * n! * (1 - 1/n + 3/(2*n^2) - 2/(3*n^3) - 47/(24*n^4) + 49/(120*n^5) + 6421/(720*n^6) + ...). - _Vaclav Kotesovec_, Dec 11 2021
%t a[n_] := Sum[k! * Binomial[k + 1, n - k], {k, Floor[n/2], n}]; Array[a, 22, 0] (* _Amiram Eldar_, Nov 30 2021 *)
%o (PARI) a(n) = sum(k=n\2, n, k!*binomial(k+1, n-k));
%o (PARI) a(n) = if(n<3, 2^n, (n-2)*a(n-1)+2*(n-1)*a(n-2)+(n-2)*a(n-3));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k*(1+x)^(k+1)))
%Y Cf. A184185, A240172.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Nov 30 2021
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