A304330 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.
1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480, 111567456000, 1264085222400, 6102480384000, 10461394944000
Offset: 0
Examples
Triangle starts: [0] 1; [1] 0, 1; [2] 0, 1, 12; [3] 0, 1, 60, 360; [4] 0, 1, 252, 5040, 20160; [5] 0, 1, 1020, 52920, 604800, 1814400; [6] 0, 1, 4092, 506880, 12640320, 99792000, 239500800; [7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;
Links
- José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024.
Crossrefs
Programs
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Maple
T := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k): for n from 0 to 8 do seq(T(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)); \\ Michel Marcus, Aug 03 2025