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A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).

%I A351703 #33 May 13 2022 18:20:36
%S A351703 1,1,1,1,0,4,1,0,1,21,1,0,0,3,148,1,0,0,1,12,1305,1,0,0,0,4,70,13806,
%T A351703 1,0,0,0,1,10,465,170401,1,0,0,0,0,5,40,3591,2403640,1,0,0,0,0,1,15,
%U A351703 315,31948,38143377,1,0,0,0,0,0,6,35,2296,319068,672552730
%N A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).
%F A351703 T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * T(j,k) for n > 0.
%F A351703 T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - _Seiichi Manyama_, May 13 2022
%e A351703 Square array begins:
%e A351703       1,   1,  1,  1, 1, 1, ...
%e A351703       1,   0,  0,  0, 0, 0, ...
%e A351703       4,   1,  0,  0, 0, 0, ...
%e A351703      21,   3,  1,  0, 0, 0, ...
%e A351703     148,  12,  4,  1, 0, 0, ...
%e A351703    1305,  70, 10,  5, 1, 0, ...
%e A351703   13806, 465, 40, 15, 6, 1, ...
%o A351703 (PARI) T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
%o A351703 (PARI) T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ _Seiichi Manyama_, May 13 2022
%Y A351703 Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.
%Y A351703 Cf. A143398, A351761.
%K A351703 nonn,tabl
%O A351703 0,6
%A A351703 _Seiichi Manyama_, Feb 20 2022