(MAGMAMagma) [[(-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

proposed

approved

editing

proposed

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, 468, 144, 9, 1

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

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

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

Triangle begins as

1;

1, 1;

4, 2, 1;

13, 12, 3, 1;

46, 52, 24, 4, 1;

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

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

(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

(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

approved

editing

_Philippe DELEHAM_, Deléham_, Dec 13 2009

_Philippe DELEHAM (kolotoko(AT)wanadoo.fr), _, Dec 13 2009

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, 468, 9, 1

0,4

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.

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

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 13 2009

approved