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%I #6 Sep 08 2022 08:45:52
%S 1,1,4,1,9,42,1,16,130,680,1,25,315,2555,14630,1,36,651,7616,63126,
%T 389592,1,49,1204,19236,219450,1871562,12314148,1,64,2052,42960,
%U 647130,7346592,64578228,449324304,1,81,3285,87285,1679535,24557247,280146867,2537661555,18555052230
%N Triangle, read by rows, T(n, k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)!*j!).
%C Row sums are: {1, 5, 52, 827, 17526, 461022, 14425650, 521941331, 21399188086, 979196554118, 49420150452256, ...}.
%H G. C. Greubel, <a href="/A176080/b176080.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)!*j!).
%F T(n, k) = binomial(n+k,n)*2F0(-n, -k; -; 1), where 2F0 is a hypergeometric function. - _G. C. Greubel_, Nov 27 2019
%e Triangle begins as:
%e 1;
%e 1, 4;
%e 1, 9, 42;
%e 1, 16, 130, 680;
%e 1, 25, 315, 2555, 14630;
%e 1, 36, 651, 7616, 63126, 389592;
%e 1, 49, 1204, 19236, 219450, 1871562, 12314148;
%e 1, 64, 2052, 42960, 647130, 7346592, 64578228, 449324304;
%p b:=binomial; T(n,k):=b(n+k,n)*add(j!*b(n,j)*b(k,j), j=0..k); seq(seq(T(n,k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 27 2019
%t T[n_, k_]:= Sum[(n+k)!/((n-j)!*(k-j)!*j!), {j,0,k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
%t Table[Binomial[n+k, n]*HypergeometricPFQ[{-n, -k}, {}, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 27 2019 *)
%o (PARI) b=binomial; T(n,k) = b(n+k,n)*sum(j=0,k, j!*b(n,j)*b(k,j)); \\ _G. C. Greubel_, Nov 27 2019
%o (Magma) B:=Binomial; [B(n+k,n)*(&+[Factorial(j)*B(n,j)*B(k,j): j in [0..k]]) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 27 2019
%o (Sage) b=binomial; [[b(n+k,n)*sum(factorial(j)*b(n,j)*b(k,j) for j in (0..k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 27 2019
%o (GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> B(n+k,n)*Sum( [0..k], j-> Factorial(j)*B(n,j)*B(k,j)) ))); # _G. C. Greubel_, Nov 27 2019
%K nonn,tabl
%O 0,3
%A _Roger L. Bagula_, Apr 08 2010
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