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A350594 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..2*n} (-1)^(n+j) * binomial(2*n,j)^k.

%I A350594 #20 Jan 08 2022 22:44:10
%S A350594 1,1,-1,1,0,1,1,2,0,-1,1,6,6,0,1,1,14,90,20,0,-1,1,30,786,1680,70,0,1,
%T A350594 1,62,5730,61340,34650,252,0,-1,1,126,38466,1696800,5562130,756756,
%U A350594 924,0,1,1,254,247170,41312060,613591650,549676764,17153136,3432,0,-1
%N A350594 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{j=0..2*n} (-1)^(n+j) * binomial(2*n,j)^k.
%H A350594 Seiichi Manyama, <a href="/A350594/b350594.txt">Antidiagonals n = 0..50, flattened</a>
%e A350594 Square array begins:
%e A350594    1, 1,   1,      1,         1,            1, ...
%e A350594   -1, 0,   2,      6,        14,           30, ...
%e A350594    1, 0,   6,     90,       786,         5730, ...
%e A350594   -1, 0,  20,   1680,     61340,      1696800, ...
%e A350594    1, 0,  70,  34650,   5562130,    613591650, ...
%e A350594   -1, 0, 252, 756756, 549676764, 248832363780, ...
%o A350594 (PARI) T(n, k) = sum(j=0, 2*n, (-1)^(n+j)*binomial(2*n, j)^k);
%Y A350594 Columns k=0..6 give A033999, A000007, A000984, A006480, A050983, A050984, A227357.
%Y A350594 Rows n=0..1 give A000012, A000918.
%Y A350594 Main diagonal gives A350595.
%Y A350594 Cf. A257050, A309010.
%K A350594 sign,tabl
%O A350594 0,8
%A A350594 _Seiichi Manyama_, Jan 08 2022