A346550 Expansion of Sum_{k>=0} k! * x^k * (1 + x)^(k+1).

1, 2, 4, 13, 54, 278, 1704, 12126, 98280, 893904, 9017280, 99918120, 1206500400, 15768729360, 221792780160, 3340515069360, 53641756586880, 914849722725120, 16514863528665600, 314599179867396480, 6306817346711481600, 132727279189258656000

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..449

Formula

a(n) = Sum_{k=floor(n/2)..n} k! * binomial(k+1,n-k).

a(n) = A240172(n-1) + A240172(n) for n > 0.

a(n) = (n-2) * a(n-1) + 2 * (n-1) * a(n-2) + (n-2) * a(n-3) for n > 2.

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

Mathematica

a[n_] := Sum[k! * Binomial[k + 1, n - k], {k, Floor[n/2], n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 30 2021 *)

Prog

(PARI) a(n) = sum(k=n\2, n, k!*binomial(k+1, n-k));
(PARI) a(n) = if(n<3, 2^n, (n-2)*a(n-1)+2*(n-1)*a(n-2)+(n-2)*a(n-3));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k*(1+x)^(k+1)))

Crossrefs

Cf. A184185, A240172.

Sequence in context: A030939 A030886 A030810 * A058135 A030968 A030862

Adjacent sequences: A346547 A346548 A346549 * A346551 A346552 A346553

Keyword

nonn

Author

Seiichi Manyama, Nov 30 2021

Status

approved