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%I #35 May 25 2019 10:09:38
%S 1,0,1,0,-1,1,0,1,0,1,0,-1,1,-1,1,0,1,5,5,0,1,0,-1,36,-120,15,-1,1,0,
%T 1,329,6286,2380,56,0,1,0,-1,3655,-557991,1056496,-52556,203,-1,1,0,1,
%U 47844,74741031,1006985994,197741887,1192625,757,0,1
%N A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.
%H Seiichi Manyama, <a href="/A308356/b308356.txt">Antidiagonals n = 0..50, flattened</a>
%F A(n,k) = Sum_{i=k..k*n} b(i) where Sum_{i=k..k*n} b(i) * (-x)^i/i! = (1/k!) * (Sum_{i=1..n} x^i/i!)^k.
%e For (n,k) = (3,2), (1/2) * (Sum_{i=1..3} x^i/i!)^2 = (1/2) * (x + x^2/2 + x^3/6)^2 = (-x)^2/2 + (-3)*(-x)^3/6 + 7*(-x)^4/24 + (-10)*(-x)^5/120 + 10*(-x)^6/720. So A(3,2) = 1 - 3 + 7 - 10 + 10 = 5.
%e Square array begins:
%e 1, 0, 0, 0, 0, 0, ...
%e 1, -1, 1, -1, 1, -1, ...
%e 1, 0, 1, 5, 36, 329, ...
%e 1, -1, 5, -120, 6286, -557991, ...
%e 1, 0, 15, 2380, 1056496, 1006985994, ...
%e 1, -1, 56, -52556, 197741887, -2063348839223, ...
%e 1, 0, 203, 1192625, 38987482590, 4546553764660831, ...
%Y Columns k=0..4 give A000012, (-1)*A000035, A307349, (-1)*A307350, A307351.
%Y Rows n=0..5 give A000007, A033999, A278990, A308363, A308389, A308390.
%Y Main diagonal gives A308327.
%Y Cf. A144510.
%K sign,tabl
%O 0,18
%A _Seiichi Manyama_, May 21 2019
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