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%I #14 Sep 08 2022 08:45:12
%S 1,4,14,52,217,1040,5768,36992,272584,2285184,21550656,226071744,
%T 2611146384,32911082496,449243785728,6598780563456,103734755882496,
%U 1737181702840320,30866291090657280,579859321408266240
%N Fifth column (k=4) of triangle A084938.
%H G. C. Greubel, <a href="/A090319/b090319.txt">Table of n, a(n) for n = 0..150</a>
%F a(n) = Sum_{k=0..n} A090595(k)*(n-k)!.
%F a(n) = Sum_{a+b+c+d = n} a!*b!*c!*d!.
%F a(n) = Sum_{k=0..n} A003149(k)*A003149(n-k).
%F G.f.: (Sum_{k>=0} k!*x^k)^4.
%F From _G. C. Greubel_, Dec 29 2019: (Start)
%F a(n) = (n+3)!*Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} Beta(k+3, n-k+1)*Beta(m+2, k-m+1)*Beta(j+1, m-j+1), where Beta(x,y) is the Beta function.
%F a(n) = Sum_{k=0..n} Sum_{m=0..k} Sum_{j=0..m} n!/(binomial(n,k) * binomial(k,m) * binomial(m,j)). (End)
%p seq( (n+3)!*add(add(add( Beta(k+3,n-k+1)*Beta(m+2,k-m+1)*Beta(j+1,m-j+1), j=0..m), m=0..k), k=0..n), n=0..20); # _G. C. Greubel_, Dec 29 2019
%t Table[(n+3)!*Sum[Beta[k+3, n-k+1]*Beta[m+2, k-m+1]*Beta[j+1, m-j+1], {k,0,n}, {m,0,k}, {j,0,m}], {n,0,20}] (* _G. C. Greubel_, Dec 29 2019 *)
%o (PARI) vector(21, n, my(b=binomial); sum(k=0,n-1, sum(m=0,k, sum(j=0,m, (n-1)!/(b(n-1,k)*b(k,m)*b(m,j)) )))) \\ _G. C. Greubel_, Dec 29 2019
%o (Magma) F:=Factorial; B:=Binomial; [ (&+[(&+[(&+[F(n)/(B(n,k)*B(k,m)*B(m,j)): j in [0..m]]): m in [0..k]]): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Dec 29 2019
%o (Sage) b=binomial; [sum(sum(sum(factorial(n)/(b(n,k)*b(k,m)*b(m,j)) for j in (0..m)) for m 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], m-> Sum([0..m], j-> Factorial(n)/(B(n,k)*B(k,m)*B(m,j)) )))); # _G. C. Greubel_, Dec 29 2019
%Y Cf. A084938.
%Y Columns, for k = 0, 1, 2, 3 : A000007, A000142, A003149, A090595.
%K easy,nonn
%O 0,2
%A _Philippe Deléham_, Feb 05 2004
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