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%I #4 Oct 08 2018 18:15:29
%S 1,1,0,1,1,0,1,3,3,0,1,6,23,13,0,1,10,86,261,75,0,1,15,230,1836,3947,
%T 541,0,1,21,505,7900,52250,74613,4683,0,1,28,973,25425,361754,1858716,
%U 1692563,47293,0,1,36,1708,67473,1706629,20706700,79345346,44794221,545835,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k - Sum_{j=1..k} exp(j*x)).
%F E.g.f. of column k: 1/(1 + k - exp(x)*(exp(k*x) - 1)/(exp(x) - 1)).
%e E.g.f. of column k: A_k(x) = 1 + (1/2)*k*(k + 1)*x/1! + (1/6)*k*(3*k^3 + 8*k^2 + 6*k + 1)*x^2/2! + (1/4)*k^2*(k + 1)^2*(3*k^2 + 7*k + 3)*x^3/3! + (1/30)*k*(45*k^7 + 270*k^6 + 635*k^5 + 741*k^4 + 440*k^3 + 115*k^2 + 5*k - 1)*x^4/4! + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 3, 6, 10, 15, ...
%e 0, 3, 23, 86, 230, 505, ...
%e 0, 13, 261, 1836, 7900, 25425, ...
%e 0, 75, 3947, 52250, 361754, 1706629, ...
%e 0, 541, 74613, 1858716, 20706700, 143195025, ...
%t Table[Function[k, n! SeriesCoefficient[1/(1 + k - Sum[Exp[i x], {i, 1, k}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
%t Table[Function[k, n! SeriesCoefficient[1/(1 + k - Exp[x] (Exp[k x] - 1)/(Exp[x] - 1)), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
%Y Columns k=0..10 give A000007, A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708.
%Y Main diagonal gives A319508.
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, Oct 08 2018
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