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%I #9 Sep 08 2022 08:45:52
%S 1,1,3,1,5,13,1,7,28,73,1,9,49,185,501,1,11,76,381,1426,4051,1,13,109,
%T 685,3331,12607,37633,1,15,148,1121,6756,32593,125882,394353,1,17,193,
%U 1713,12361,73129,354033,1401409,4596553,1,19,244,2485,20926,147295,865936,4233673,17209234,58941091
%N Triangle, read by rows, T(n,k) = Sum_{j=1..k} binomial(n-1, j-1)*binomial(k, j - 1)*(j-1)!.
%C Row sums are: {1, 4, 19, 109, 745, 5946, 54379, 560869, 6439409, 81420904, ...}.
%H G. C. Greubel, <a href="/A176122/b176122.txt">Rows n = 1..100 of triangle, flattened</a>
%F T(n,k) = Sum_{j=1..k} binomial(n-1, j-1)*binomial(k, j - 1)*(j-1)!.
%e Triangle begins as:
%e 1;
%e 1, 3;
%e 1, 5, 13;
%e 1, 7, 28, 73;
%e 1, 9, 49, 185, 501;
%e 1, 11, 76, 381, 1426, 4051;
%e 1, 13, 109, 685, 3331, 12607, 37633;
%e 1, 15, 148, 1121, 6756, 32593, 125882, 394353;
%e 1, 17, 193, 1713, 12361, 73129, 354033, 1401409, 4596553;
%e 1, 19, 244, 2485, 20926, 147295, 865936, 4233673, 17209234, 58941091;
%p b:=binomial; T(n,k):=add((j-1)!*b(n-1,j-1)*b(k,j-1), j=1..k); seq(seq(T(n,k), k=1..n), n=1..10); # _G. C. Greubel_, Nov 27 2019
%t T[n_, k_]:= Sum[Binomial[n-1, j-1]*Binomial[k, j-1]*(j-1)!, {j,k}]; Table[T[n, k], {n,10},{k,n}]//Flatten
%o (PARI) b=binomial; T(n,k) = sum(j=1,k, (j-1)!*b(n-1,j-1)*b(k,j-1)); \\ _G. C. Greubel_, Nov 27 2019
%o (Magma) B:=Binomial; [(&+[Factorial(j-1)*B(n-1,j-1)*B(k,j-1): j in [1..k]]) : k in [1..n], n in [1..10]]; // _G. C. Greubel_, Nov 27 2019
%o (Sage) b=binomial; [[sum(factorial(j-1)*b(n-1,j-1)*b(k,j-1) for j in (1..k)) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Nov 27 2019
%o (GAP) B:=Binomial;; Flat(List([1..10], n-> List([1..n], k-> Sum([0..k], j-> Factorial(j-1)*B(n-1,j-1)*B(k,j-1)) ))); # _G. C. Greubel_, Nov 27 2019
%K nonn,tabl
%O 1,3
%A _Roger L. Bagula_, Apr 09 2010
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