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A355666 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k/k! * (exp(x) - 1)).

%I A355666 #18 Sep 02 2022 12:07:37
%S A355666 1,1,1,1,0,3,1,0,2,13,1,0,0,3,75,1,0,0,3,28,541,1,0,0,0,6,125,4683,1,
%T A355666 0,0,0,4,10,1146,47293,1,0,0,0,0,10,195,8827,545835,1,0,0,0,0,5,20,
%U A355666 1281,94200,7087261,1,0,0,0,0,0,15,35,5908,1007001,102247563,1,0,0,0,0,0,6,35,1176,68076,12814390,1622632573
%N A355666 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k/k! * (exp(x) - 1)).
%F A355666 T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=k+1..n} binomial(n-k,j-k) * T(n-j,k) for n > 0.
%F A355666 T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
%e A355666 Square array begins:
%e A355666      1,    1,   1,  1,  1, 1, 1, ...
%e A355666      1,    0,   0,  0,  0, 0, 0, ...
%e A355666      3,    2,   0,  0,  0, 0, 0, ...
%e A355666     13,    3,   3,  0,  0, 0, 0, ...
%e A355666     75,   28,   6,  4,  0, 0, 0, ...
%e A355666    541,  125,  10, 10,  5, 0, 0, ...
%e A355666   4683, 1146, 195, 20, 15, 6, 0, ...
%o A355666 (PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));
%Y A355666 Columns k=0..3 give A000670, A052848, A353998, A353999.
%Y A355666 Cf. A351703, A355652.
%K A355666 nonn,tabl
%O A355666 0,6
%A A355666 _Seiichi Manyama_, Jul 13 2022