|
|
A090595
|
|
Fourth column (k=3) of triangle A084938.
|
|
3
|
|
|
1, 3, 9, 31, 126, 606, 3428, 22572, 170856, 1467432, 14123808, 150644448, 1763377344, 22466496960, 309371685120, 4577183527680, 72390548206080, 1218507923427840, 21746087150745600, 410094720409651200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} A003149(k)*(n-k)!.
G.f.: (Sum_k>=0} k!*x^k)^3.
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
|
|
|
STATUS
|
approved
|
|
|
|