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Triangle, read by rows, T(n, k) = Sum_{j=0..k} (-1)^j*(k-j+1)^n*binomial(n+1, j) *binomial(n+2, j)/(j+1).
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%I #6 Sep 08 2022 08:45:52

%S 1,1,-1,1,-2,-9,1,-2,-33,-56,1,1,-109,-209,50,1,11,-324,-894,1641,

%T 10485,1,36,-867,-4262,12951,85926,211435,1,92,-2085,-20516,74369,

%U 625164,1435939,1005536,1,211,-4419,-93989,344617,4306671,9337441,-7280909,-94801266

%N Triangle, read by rows, T(n, k) = Sum_{j=0..k} (-1)^j*(k-j+1)^n*binomial(n+1, j) *binomial(n+2, j)/(j+1).

%C Row sums are: {1, 0, -10, -90, -266, 10920, 305220, 3118500, -88191642, -5436127620, -116211623892, ...}.

%H G. C. Greubel, <a href="/A176124/b176124.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = Sum_{j=0..k} (-1)^j*(k-j+1)^n*binomial(n+1,j)*binomial(n+2,j)/(j+1).

%e Triangle begins as:

%e 1;

%e 1, -1; 1, -2, -9;

%e 1, -2, -33, -56;

%e 1, 1, -109, -209, 50;

%e 1, 11, -324, -894, 1641, 10485;

%e 1, 36, -867, -4262, 12951, 85926, 211435;

%e 1, 92, -2085, -20516, 74369, 625164, 1435939, 1005536;

%p b:=binomial; T(n,k):=add((-1)^j*(k-j+1)^n*b(n+1,j)*b(n+2,j)/(j+1), 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[(-1)^j*(k-j+1)^n*Binomial[n+1,j]*Binomial[n+2,j]/(j+1), {j, 0, k}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (PARI) b=binomial; T(n,k) = sum(j=0,k, (-1)^j*(k-j+1)^n*b(n,j)*b(k,j)/(j+1)); \\ _G. C. Greubel_, Nov 27 2019

%o (Magma) B:=Binomial; [(&+[(-1)^j*(k-j+1)^n*B(n+1,j)*B(n+2,j)/(j+1): j in [0..k]]) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 27 2019

%o (Sage) b=binomial; [[sum((-1)^j*(k-j+1)^n*b(n+1,j)*b(n+2,j)/(j+1) 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-> Sum([0..k], j-> (-1)^j*(k-j+1)^n*B(n+1,j)*B(n+2,j)/(j+1)) ))); # _G. C. Greubel_, Nov 27 2019

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 09 2010