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

Author

Peter Luschny, Nov 07 2024

Keywords

• nonn

Crossrefs

Cf. A377661, A073107.

Programs

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}]

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).

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)