A304334 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.
1, 0, 1, 0, 1, 6, 0, 1, 30, 60, 0, 1, 126, 840, 840, 0, 1, 510, 8820, 25200, 15120, 0, 1, 2046, 84480, 526680, 831600, 332640, 0, 1, 8190, 780780, 9609600, 30270240, 30270240, 8648640, 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200
Offset: 0
Examples
Triangle starts: [0] 1 [1] 0, 1 [2] 0, 1, 6 [3] 0, 1, 30, 60 [4] 0, 1, 126, 840, 840 [5] 0, 1, 510, 8820, 25200, 15120 [6] 0, 1, 2046, 84480, 526680, 831600, 332640 [7] 0, 1, 8190, 780780, 9609600, 30270240, 30270240, 8648640 [8] 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200
Crossrefs
Programs
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Maple
A304334 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/k!: for n from 0 to 8 do seq(A304334(n, k), k=0..n) od;
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PARI
T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/k!; tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 11 2018
Formula
T(n, k) = A304330(n, k) / k!.
Comments