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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).
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%I #22 Mar 07 2023 11:11:59

%S 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 11,85,529,2261,4051,1,1,1,13,121,961,6121,24781,37633,1,1,1,15,163,

%U 1561,13041,82711,309835,394353,1,1,1,17,211,2353,24101,207001,1273567,4342241,4596553,1

%N 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 Seiichi Manyama, <a href="/A293012/b293012.txt">Antidiagonals n = 0..139, flattened</a>

%F E.g.f. of column k: exp(x/(1 - x)^k).

%F From _Seiichi Manyama_, Oct 21 2017: (Start)

%F Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

%F 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 A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - _Seiichi Manyama_, Mar 06 2023

%e 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 Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 3, 5, 7, 9, 11, ...

%e 1, 13, 31, 55, 85, 121, ...

%e 1, 73, 241, 529, 961, 1561, ...

%e 1, 501, 2261, 6121, 13041, 24101, ...

%t Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

%o (PARI) T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ _Seiichi Manyama_, Mar 06 2023

%Y Columns k=0..4 give A000012, A000262, A082579, A091695, A361283.

%Y Main diagonal gives A293013.

%K nonn,tabl

%O 0,9

%A _Ilya Gutkovskiy_, Sep 28 2017