A129833 a(n) = Sum_{k = 0..n } binomial(n + 1, k + 1)*binomial(n, k)*k!.

1, 3, 11, 52, 309, 2221, 18703, 180216, 1952457, 23466223, 309577971, 4444537868, 68948023741, 1148825560377, 20455144724407, 387479309532976, 7778881684953873, 164942847995071611, 3682885668837002587, 86359724102207331876, 2121535102985378053061, 54482075844410029721893, 1459677302947807284662751

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..443

F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.

Formula

Conjecture: +(n+1)*a(n) -n*(2*n+5)*a(n-1) +(n-1)*(n^2+6*n+3)*a(n-2) -(n-2)*(3*n^2-2)*a(n-3) +(n-2)*(n-3)*(3*n-4)*a(n-4) -(n-4)*(n-3)^2*a(n-5) = 0. - R. J. Mathar, Feb 28 2015

Conjecture: (n+1)*(n^2-4*n+2)*a(n) -n*(2*n^3-5*n^2-6*n+3)*a(n-1) +n*(n-1)*(n^3-2*n^2-2*n-2)*a(n-2) -(n-2)*(n^2-2*n-1)*(n-1)^2*a(n-3) = 0. - R. J. Mathar, Feb 28 2015

a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 1/4) / sqrt(2) * (1 + 79/(48*sqrt(n))). - Vaclav Kotesovec, Oct 12 2016

From G. C. Greubel, Mar 10 2021: (Start)

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ((n+1)*k!/(k+1)).

a(n) = (n+1)*Hypergeometric3F1([-n, -n, 1], [2], 1). (End)

Maple

A129833 := proc(n)
    add(A176120(n, k), k=0..n) ;
end proc: # R. J. Mathar, Feb 28 2015

Mathematica

a[n_]:= Sum[Binomial[n+1, k+1]*Binomial[n, k]*k!, {k, 0, n}]; Table[a[n], {n, 0, 30}]

Prog

(Sage) [sum( binomial(n, k)^2*((n+1)*factorial(k)/(k+1)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Mar 10 2021
(Magma) [(&+[Binomial(n, k)^2*((n+1)*Factorial(k)/(k+1)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 10 2021
(PARI) a(n) = sum(k= 0, n, binomial(n+1, k+1)*binomial(n, k)*k!); \\ Michel Marcus, Mar 10 2021

Crossrefs

Cf. A052852, A176120.

Sequence in context: A058799 A357833 A054362 * A321585 A368283 A107958

Adjacent sequences: A129830 A129831 A129832 * A129834 A129835 A129836

Keyword

nonn,easy

Author

Roger L. Bagula, May 21 2007

Extensions

Edited by N. J. A. Sloane, Sep 30 2007

Status

approved