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A320080 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*log(1 + x)).

%I A320080 #20 Sep 11 2023 10:02:19
%S A320080 1,1,0,1,1,0,1,2,1,0,1,3,6,2,0,1,4,15,28,4,0,1,5,28,114,172,14,0,1,6,
%T A320080 45,296,1152,1328,38,0,1,7,66,610,4168,14562,12272,216,0,1,8,91,1092,
%U A320080 11020,73376,220842,132480,600,0,1,9,120,1778,24084,248870,1550048,3907656,1633344,6240,0
%N A320080 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*log(1 + x)).
%F A320080 E.g.f. of column k: 1/(1 - k*log(1 + x)).
%F A320080 A(n,k) = Sum_{j=0..n} Stirling1(n,j)*j!*k^j.
%F A320080 A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - _Seiichi Manyama_, May 22 2022
%e A320080 E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k - 1)*x^2/2! + 2*k*(3*k^2 - 3*k + 1)*x^3/3! + 2*k*(12*k^3 - 18*k^2 + 11*k - 3)*x^4/4! + ...
%e A320080 Square array begins:
%e A320080   1,   1,     1,      1,      1,       1,  ...
%e A320080   0,   1,     2,      3,      4,       5,  ...
%e A320080   0,   1,     6,     15,     28,      45,  ...
%e A320080   0,   2,    28,    114,    296,     610,  ...
%e A320080   0,   4,   172,   1152,   4168,   11020,  ...
%e A320080   0,  14,  1328,  14562,  73376,  248870,  ...
%t A320080 Table[Function[k, n! SeriesCoefficient[1/(1 - k Log[1 + x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y A320080 Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.
%Y A320080 Main diagonal gives A317172.
%Y A320080 Cf. A048594, A048994, A094416, A320079, A334369.
%K A320080 nonn,tabl
%O A320080 0,8
%A A320080 _Ilya Gutkovskiy_, Oct 05 2018