OFFSET
0,2
COMMENTS
3rd column (k=2): A003149.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
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)
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Philippe Deléham, Feb 01 2004
STATUS
approved