A090595 Fourth column (k=3) of triangle A084938.

1, 3, 9, 31, 126, 606, 3428, 22572, 170856, 1467432, 14123808, 150644448, 1763377344, 22466496960, 309371685120, 4577183527680, 72390548206080, 1218507923427840, 21746087150745600, 410094720409651200

Offset: 0

Author

Philippe Deléham, Feb 01 2004

Keywords

• easy
• nonn

Comments

3rd column (k=2): A003149.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Programs

GAP

B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], j-> Factorial(n)/(B(n,k)*B(k,j)) ))); # G. C. Greubel, Dec 29 2019

Magma

F:=Factorial; B:=Binomial; [ (&+[ (&+[F(n)/(B(k,j)*B(n,k)): j in [0..k]]) : k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

Maple

seq(factorial(n+2)*add(add(Beta(k+2, n-k+1)*Beta(j+1, k-j+1), j=0..k), k=0..n), n = 0..20); # G. C. Greubel, Dec 29 2019

Mathematica

Table[(n+2)!*Sum[Beta[k+2, n-k+1]*Beta[j+1, k-j+1], {k,0,n}, {j,0,k}], {n,0,20}] (* G. C. Greubel, Dec 29 2019 *)

PARI

vector(21, n, my(b=binomial); sum(k=0,n-1, sum(j=0,k, (n-1)!/(b(k,j)* b(n-1, k)) ))) \\ G. C. Greubel, Dec 29 2019

Sage

[ factorial(n+2)*sum(sum(beta(k+2,n-k+1)*beta(j+1,k-j+1) for j in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

Formula

a(n) = Sum_{k=0..n} A003149(k)*(n-k)!.

G.f.: (Sum_{k>=0} k!*x^k)^3.

a(n) ~ 3 * n!. - Vaclav Kotesovec, Jun 25 2019

From G. C. Greubel, Dec 29 2019: (Start)

a(n) = (n+2)!*Sum_{k=0..n} Sum_{j=0..n} B(k+2, n-k+1)*B(j+1,k-j+1), where B(x,y) is the Beta function.

a(n) = Sum_{k=0..n} Sum_{j=0..k} n!/(binomial(n,k)*binomial(k,j)). (End)