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%I #30 May 25 2019 16:29:20
%S 1,1,1,1,0,1,1,1,1,1,1,-2,3,0,1,1,9,37,9,1,1,1,-44,997,-692,31,0,1,1,
%T 265,44121,148041,14371,111,1,1,1,-1854,2882071,-66211704,25413205,
%U -315002,407,0,1,1,14833,260415373,53414037505,120965241901,4744544613,7156969,1513,1,1
%N A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..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="/A308322/b308322.txt">Antidiagonals n = 0..50, flattened</a>
%F A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * (-x)^i/i! = (Sum_{i=0..n} x^i/i!)^k.
%e For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + (-2)*(-x) + 4*(-x)^2/2 + (-8)*(-x)^3/6 + 14*(-x)^4/24 + (-20)*(-x)^5/120 + 20*(-x)^6/720. So A(3,2) = 1 - 2 + 4 - 8 + 14 - 20 + 20 = 9.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 0, 1, -2, 9, -44, ...
%e 1, 1, 3, 37, 997, 44121, ...
%e 1, 0, 9, -692, 148041, -66211704, ...
%e 1, 1, 31, 14371, 25413205, 120965241901, ...
%e 1, 0, 111, -315002, 4744544613, -247578134832564, ...
%e 1, 1, 407, 7156969, 935728207597, 545591130328772081, ...
%Y Columns k=0..5 give A000012, A059841, A120305, A307318, A307324, A308325.
%Y Rows n=0..1 give A000012, A182386.
%Y Main diagonal gives A308323.
%Y Cf. A308292.
%K sign,tabl
%O 0,12
%A _Seiichi Manyama_, May 20 2019
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