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%I #10 Sep 08 2022 08:45:52
%S 1,1,6,1,12,70,1,20,195,1190,1,30,441,4088,26334,1,42,868,11424,
%T 105210,714252,1,56,1548,27480,344850,3208392,22869132,1,72,2565,
%U 59070,970695,11938212,113011899,842483070,1,90,4015,116380,2425995,38200162,466911445,4511398320,35048431990
%N Triangle, read by rows, T(n,k) = binomial(n+k+1, n+1) * Sum_{j=0..k} j!*binomial(n,j)*binomial(k, j).
%C Row sums are: {1, 7, 83, 1406, 30894, 831797, 26451459, 968465584, 40067488398, 1846559924606, 93734743085246, ...}.
%H G. C. Greubel, <a href="/A176121/b176121.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = binomial(n+k+1, n+1) * Sum_{j=0..k} j!*binomial(n,j)*binomial(k, j).
%e Triangle begins as:
%e 1;
%e 1, 6;
%e 1, 12, 70;
%e 1, 20, 195, 1190;
%e 1, 30, 441, 4088, 26334;
%e 1, 42, 868, 11424, 105210, 714252;
%e 1, 56, 1548, 27480, 344850, 3208392, 22869132;
%e 1, 72, 2565, 59070, 970695, 11938212, 113011899, 842483070;
%e ...
%p b:=binomial; T(n,k):=b(n+k+1,n+1)*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_] = Binomial[n+k+1,n+1]*Sum[Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o (PARI) b=binomial; T(n,k) = b(n+k+1,n+1)*sum(j=0,k, j!*b(n,j)*b(k,j)); \\ _G. C. Greubel_, Nov 27 2019
%o (Magma) B:=Binomial; [B(n+k+1,n+1)*(&+[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+1,n+1)*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+1,n+1)*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 09 2010
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