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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.

%I A304334 #15 May 12 2018 07:10:44
%S A304334 1,0,1,0,1,6,0,1,30,60,0,1,126,840,840,0,1,510,8820,25200,15120,0,1,
%T A304334 2046,84480,526680,831600,332640,0,1,8190,780780,9609600,30270240,
%U A304334 30270240,8648640,0,1,32766,7108920,164684520,929728800,1755673920,1210809600,259459200
%N 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.
%F A304334 T(n, k) = A304330(n, k) / k!.
%e A304334 Triangle starts:
%e A304334 [0] 1
%e A304334 [1] 0, 1
%e A304334 [2] 0, 1,     6
%e A304334 [3] 0, 1,    30,      60
%e A304334 [4] 0, 1,   126,     840,       840
%e A304334 [5] 0, 1,   510,    8820,     25200,     15120
%e A304334 [6] 0, 1,  2046,   84480,    526680,    831600,     332640
%e A304334 [7] 0, 1,  8190,  780780,   9609600,  30270240,   30270240,    8648640
%e A304334 [8] 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200
%p A304334 A304334 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/k!:
%p A304334 for n from 0 to 8 do seq(A304334(n, k), k=0..n) od;
%o A304334 (PARI) T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/k!;
%o A304334 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 11 2018
%Y A304334 Row sums are bisection of A081562, T(n,n) ~ A000407, T(n,n-1) ~ A048854(n,2), T(n,2) ~ A002446.
%Y A304334 Cf. A304330, A304336.
%K A304334 nonn,tabl
%O A304334 0,6
%A A304334 _Peter Luschny_, May 11 2018