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A384801 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A213108.

%I A384801 #15 Jul 27 2025 17:59:49
%S A384801 1,1,0,1,1,0,1,2,3,0,1,3,8,10,0,1,4,15,38,41,0,1,5,24,90,216,76,0,1,6,
%T A384801 35,172,633,1162,-2183,0,1,7,48,290,1424,4668,2236,-54998,0,1,8,63,
%U A384801 450,2745,12724,30177,-102282,-1045567,0,1,9,80,658,4776,28300,113080,43914,-3135056,-15948296,0
%N A384801 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A213108.
%F A384801 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).
%e A384801 Square array begins:
%e A384801   1,     1,    1,     1,      1,      1,      1, ...
%e A384801   0,     1,    2,     3,      4,      5,      6, ...
%e A384801   0,     3,    8,    15,     24,     35,     48, ...
%e A384801   0,    10,   38,    90,    172,    290,    450, ...
%e A384801   0,    41,  216,   633,   1424,   2745,   4776, ...
%e A384801   0,    76, 1162,  4668,  12724,  28300,  55326, ...
%e A384801   0, -2183, 2236, 30177, 113080, 302305, 675252, ...
%o A384801 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
%o A384801 a(n, k) = b(n, -k);
%Y A384801 Columns k=0..1 give A000007, A213108.
%K A384801 sign,tabl
%O A384801 0,8
%A A384801 _Seiichi Manyama_, Jun 10 2025