A377662 a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=k..n} n!/(k!*(j - k)!). Row sums of A377661.

1, 3, 14, 80, 534, 4102, 35916, 354888, 3915750, 47754938, 637840356, 9256590928, 144977618044, 2436460447020, 43719637179224, 834042701945520, 16852447379512710, 359468276129261730, 8070500634880125300, 190211302604157871680, 4695001374741310892820

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = e Sum_{k=0..n} binomial(n, k)^2 * Gamma(n - k + 1, 1).

a(n) = Sum_{k=0..n} binomial(n, k)^2 * KummerU(k - n, k - n, 1), k = 0..n).

a(n) = Sum_{k=0..n} binomial(n, k) * A073107(n, k).

From Vaclav Kotesovec, Nov 07 2024: (Start)

Recurrence: n*(n^2 - 9*n + 12)*a(n) = 2*(n^4 - 7*n^3 - 5*n^2 + 29*n - 12)*a(n-1) - (n-1)*(n^4 - 2*n^3 - 47*n^2 + 104*n - 48)*a(n-2) + 2*(n-2)^2*(2*n - 3)*(n^2 - 7*n + 4)*a(n-3).

a(n) ~ 2^(-1/2) * exp(2*sqrt(n) - n + 1/2) * n^(n + 1/4) * (1 + 31/(48*sqrt(n))). (End)

Maple

A377662 := n -> add(binomial(n, k)*add(n!/(k!*(j-k)!), j = k..n), k = 0..n):
seq(A377662(n), n = 0..20);
# Or:
a := n -> add(binomial(n, k)^2 * KummerU(k-n, k-n, 1), k = 0..n):
seq(simplify(a(n)), n = 0..20);

Mathematica

a[n_] := E Sum[Gamma[n - k + 1, 1] Binomial[n, k]^2, {k, 0, n}];
Table[a[n], {n, 0, 20}]

Crossrefs

Cf. A377661, A073107.

Sequence in context: A212391 A000264 A009053 * A202474 A347001 A354325

Adjacent sequences: A377658 A377659 A377661 * A377663 A377664 A377665

Keyword

nonn

Author

Peter Luschny, Nov 07 2024

Status

approved