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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).
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%I #25 Jul 10 2022 08:07:41

%S 1,1,1,1,0,2,1,0,2,5,1,0,0,3,15,1,0,0,6,16,52,1,0,0,0,12,65,203,1,0,0,

%T 0,24,20,336,877,1,0,0,0,0,60,390,1897,4140,1,0,0,0,0,120,120,2562,

%U 11824,21147,1,0,0,0,0,0,360,210,11816,80145,115975,1,0,0,0,0,0,720,840,20496,105912,586000,678570

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

%H Seiichi Manyama, <a href="/A292892/b292892.txt">Antidiagonals n = 0..139, flattened</a>

%F From _Seiichi Manyama_, Jul 09 2022: (Start)

%F T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.

%F T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

%e Square array begins:

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

%e 1, 0, 0, 0, 0, ...

%e 2, 2, 0, 0, 0, ...

%e 5, 3, 6, 0, 0, ...

%e 15, 16, 12, 24, 0, ...

%e 52, 65, 20, 60, 120, ...

%o (PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ _Seiichi Manyama_, Jul 09 2022

%Y Columns k=0..3 give A000110, A052506, A240989, A292891.

%Y Rows n=0..1 give A000012, A000007.

%Y Main diagonal gives A000007.

%Y Cf. A292894, A355607.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Sep 26 2017