%I #7 Sep 08 2022 08:45:50

%S 1,1,1,4,2,1,13,12,3,1,46,52,24,4,1,166,230,130,40,5,1,610,996,690,

%T 260,60,6,1,2269,4270,3486,1610,455,84,7,1,8518,18152,17080,9296,3220,

%U 728,112,8,1,32206,76662,81684,51240,20916,5796,1092,144,9,1

%N Triangle T, read by rows : T(n,k) = A007318(n,k)*A026641(n-k).

%H G. C. Greubel, <a href="/A171650/b171650.txt">Rows n = 0..100 of triangle, flattened</a>

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = A127361(n), A127328(n), A026641(n), A126568(n), A133158(n) for x = -2, -1, 0, 1, 2 respectively.

%F T(n, k) = (-1)^(n-k)*binomial(n, k)*Sum_{j=0..n-k} (-1)^j*Binomial(n-k+j, j). - _G. C. Greubel_, Apr 29 2019

%e Triangle begins as

%e 1;

%e 1, 1;

%e 4, 2, 1;

%e 13, 12, 3, 1;

%e 46, 52, 24, 4, 1;

%e 166, 230, 130, 40, 5, 1; ...

%t T[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Sum[(-1)^j*Binomial[n-k+j, j], {j, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 29 2019 *)

%o (PARI) {T(n,k) = (-1)^(n-k)*binomial(n,k)*sum(j=0,n-k,(-1)^j*binomial(n-k+j,j))}; \\ _G. C. Greubel_, Apr 29 2019

%o (Magma) [[(-1)^(n-k)*Binomial(n,k)*(&+[(-1)^j*Binomial(n-k+j,j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Apr 29 2019

%o (Sage) [[(-1)^(n-k)*binomial(n,k)*sum((-1)^j*binomial(n-k+j,j) for j in (0..n-k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 29 2019

%K nonn,tabl

%O 0,4

%A _Philippe Deléham_, Dec 13 2009