A171650 Triangle T, read by rows : T(n,k) = A007318(n,k)*A026641(n-k).

1, 1, 1, 4, 2, 1, 13, 12, 3, 1, 46, 52, 24, 4, 1, 166, 230, 130, 40, 5, 1, 610, 996, 690, 260, 60, 6, 1, 2269, 4270, 3486, 1610, 455, 84, 7, 1, 8518, 18152, 17080, 9296, 3220, 728, 112, 8, 1, 32206, 76662, 81684, 51240, 20916, 5796, 1092, 144, 9, 1

Offset: 0

Author

Philippe Deléham, Dec 13 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as
    1;
    1,   1;
    4,   2,   1;
   13,  12,   3,  1;
   46,  52,  24,  4, 1;
  166, 230, 130, 40, 5, 1; ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

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

Mathematica

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 *)

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

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

Formula

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.

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