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%I #16 Sep 08 2022 08:45:12
%S 1,3,9,31,126,606,3428,22572,170856,1467432,14123808,150644448,
%T 1763377344,22466496960,309371685120,4577183527680,72390548206080,
%U 1218507923427840,21746087150745600,410094720409651200
%N Fourth column (k=3) of triangle A084938.
%C 3rd column (k=2): A003149.
%H Vaclav Kotesovec, <a href="/A090595/b090595.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} A003149(k)*(n-k)!.
%F G.f.: (Sum_k>=0} k!*x^k)^3.
%F a(n) ~ 3 * n!. - _Vaclav Kotesovec_, Jun 25 2019
%F From _G. C. Greubel_, Dec 29 2019: (Start)
%F 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.
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} n!/(binomial(n,k)*binomial(k,j)). (End)
%p 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
%t 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 *)
%o (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
%o (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
%o (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
%o (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
%K easy,nonn
%O 0,2
%A _Philippe Deléham_, Feb 01 2004
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