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A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

%I A293012 #22 Mar 07 2023 11:11:59
%S A293012 1,1,1,1,1,1,1,1,3,1,1,1,5,13,1,1,1,7,31,73,1,1,1,9,55,241,501,1,1,1,
%T A293012 11,85,529,2261,4051,1,1,1,13,121,961,6121,24781,37633,1,1,1,15,163,
%U A293012 1561,13041,82711,309835,394353,1,1,1,17,211,2353,24101,207001,1273567,4342241,4596553,1
%N A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).
%H A293012 Seiichi Manyama, <a href="/A293012/b293012.txt">Antidiagonals n = 0..139, flattened</a>
%F A293012 E.g.f. of column k: exp(x/(1 - x)^k).
%F A293012 From _Seiichi Manyama_, Oct 21 2017: (Start)
%F A293012 Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
%F A293012 A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0. (End)
%F A293012 A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - _Seiichi Manyama_, Mar 06 2023
%e A293012 E.g.f. of column k: A_k(x) =  1 + x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 9*k + 1)*x^3/3! + (4*k^3 + 36*k^2 + 32*k + 1)*x^4/4! + ...
%e A293012 Square array begins:
%e A293012   1,   1,    1,    1,     1,     1,  ...
%e A293012   1,   1,    1,    1,     1,     1,  ...
%e A293012   1,   3,    5,    7,     9,    11,  ...
%e A293012   1,  13,   31,   55,    85,   121,  ...
%e A293012   1,  73,  241,  529,   961,  1561,  ...
%e A293012   1, 501, 2261, 6121, 13041, 24101,  ...
%t A293012 Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%o A293012 (PARI) T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ _Seiichi Manyama_, Mar 06 2023
%Y A293012 Columns k=0..4 give A000012, A000262, A082579, A091695, A361283.
%Y A293012 Main diagonal gives A293013.
%K A293012 nonn,tabl
%O A293012 0,9
%A A293012 _Ilya Gutkovskiy_, Sep 28 2017