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A320253 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)).

%I A320253 #4 Oct 08 2018 18:15:29
%S A320253 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 A320253 541,0,1,21,505,7900,52250,74613,4683,0,1,28,973,25425,361754,1858716,
%U A320253 1692563,47293,0,1,36,1708,67473,1706629,20706700,79345346,44794221,545835,0
%N A320253 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 A320253 E.g.f. of column k: 1/(1 + k - exp(x)*(exp(k*x) - 1)/(exp(x) - 1)).
%e A320253 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 A320253 Square array begins:
%e A320253   1,    1,      1,        1,         1,          1,  ...
%e A320253   0,    1,      3,        6,        10,         15,  ...
%e A320253   0,    3,     23,       86,       230,        505,  ...
%e A320253   0,   13,    261,     1836,      7900,      25425,  ...
%e A320253   0,   75,   3947,    52250,    361754,    1706629,  ...
%e A320253   0,  541,  74613,  1858716,  20706700,  143195025,  ...
%t A320253 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 A320253 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 A320253 Columns k=0..10 give A000007, A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708.
%Y A320253 Main diagonal gives A319508.
%K A320253 nonn,tabl
%O A320253 0,8
%A A320253 _Ilya Gutkovskiy_, Oct 08 2018