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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.
8

%I #13 May 10 2024 02:42:34

%S 1,0,1,0,1,12,0,1,60,360,0,1,252,5040,20160,0,1,1020,52920,604800,

%T 1814400,0,1,4092,506880,12640320,99792000,239500800,0,1,16380,

%U 4684680,230630400,3632428800,21794572800,43589145600,0,1,65532,42653520,3952428480

%N 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.

%H José L. Cereceda, <a href="https://arxiv.org/abs/2405.05268">Sums of powers of integers and the sequence A304330</a>, arXiv:2405.05268 [math.GM], 2024.

%e Triangle starts:

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 1, 12;

%e [3] 0, 1, 60, 360;

%e [4] 0, 1, 252, 5040, 20160;

%e [5] 0, 1, 1020, 52920, 604800, 1814400;

%e [6] 0, 1, 4092, 506880, 12640320, 99792000, 239500800;

%e [7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;

%p T := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k):

%p for n from 0 to 8 do seq(T(n, k), k=0..n) od;

%Y Row sums are A100872, T(n,2) = A058896, T(n,n) = A002674, T(n,n-1)= A091032.

%Y Cf. A304334, A304336.

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, May 11 2018