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A355650 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/k! * (exp(x) - 1)).

%I A355650 #18 Jul 12 2022 14:36:19
%S A355650 1,1,1,1,0,2,1,0,2,5,1,0,0,3,15,1,0,0,3,16,52,1,0,0,0,6,65,203,1,0,0,
%T A355650 0,4,10,336,877,1,0,0,0,0,10,105,1897,4140,1,0,0,0,0,5,20,651,11824,
%U A355650 21147,1,0,0,0,0,0,15,35,2968,80145,115975,1,0,0,0,0,0,6,35,616,18936,586000,678570
%N A355650 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/k! * (exp(x) - 1)).
%F A355650 T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * Sum_{j=k+1..n} (j/(j-k)!) * T(n-j,k)/(n-j)! for n > 0.
%F A355650 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
%e A355650 Square array begins:
%e A355650     1,   1,   1,  1,  1, 1, 1, ...
%e A355650     1,   0,   0,  0,  0, 0, 0, ...
%e A355650     2,   2,   0,  0,  0, 0, 0, ...
%e A355650     5,   3,   3,  0,  0, 0, 0, ...
%e A355650    15,  16,   6,  4,  0, 0, 0, ...
%e A355650    52,  65,  10, 10,  5, 0, 0, ...
%e A355650   203, 336, 105, 20, 15, 6, 0, ...
%o A355650 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));
%Y A355650 Columns k=0..3 give A000110, A052506, A354000, A354001.
%Y A355650 Cf. A145460, A292892, A355610.
%K A355650 nonn,tabl
%O A355650 0,6
%A A355650 _Seiichi Manyama_, Jul 12 2022