cp's OEIS Frontend

A292892 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)).

%I A292892 #25 Jul 10 2022 08:07:41
%S A292892 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 A292892 0,24,20,336,877,1,0,0,0,0,60,390,1897,4140,1,0,0,0,0,120,120,2562,
%U A292892 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 A292892 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 A292892 Seiichi Manyama, <a href="/A292892/b292892.txt">Antidiagonals n = 0..139, flattened</a>
%F A292892 From _Seiichi Manyama_, Jul 09 2022: (Start)
%F A292892 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
%F A292892 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 A292892 Square array begins:
%e A292892    1,  1,  1,  1,   1, ...
%e A292892    1,  0,  0,  0,   0, ...
%e A292892    2,  2,  0,  0,   0, ...
%e A292892    5,  3,  6,  0,   0, ...
%e A292892   15, 16, 12, 24,   0, ...
%e A292892   52, 65, 20, 60, 120, ...
%o A292892 (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 A292892 Columns k=0..3 give A000110, A052506, A240989, A292891.
%Y A292892 Rows n=0..1 give A000012, A000007.
%Y A292892 Main diagonal gives A000007.
%Y A292892 Cf. A292894, A355607.
%K A292892 nonn,tabl
%O A292892 0,6
%A A292892 _Seiichi Manyama_, Sep 26 2017