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A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

%I A294212 #33 Oct 29 2017 06:41:56
%S A294212 1,1,0,1,1,0,1,1,3,0,1,1,5,13,0,1,1,5,25,73,0,1,1,5,31,193,501,0,1,1,
%T A294212 5,31,241,1601,4051,0,1,1,5,31,265,2261,16741,37633,0,1,1,5,31,265,
%U A294212 2501,25501,190345,394353,0,1,1,5,31,265,2621,29461,319915,2509025
%N A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).
%H A294212 Seiichi Manyama, <a href="/A294212/b294212.txt">Antidiagonals n = 0..139, flattened</a>
%F A294212 B(j,k) is the coefficient of Product_{i=1..k} 1/(1-x^i).
%F A294212 A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.
%e A294212 Square array B(j,k) begins:
%e A294212    1,   1,    1,    1,    1, ...
%e A294212    0,   1,    1,    1,    1, ...
%e A294212    0,   1,    2,    2,    2, ...
%e A294212    0,   1,    2,    3,    3, ...
%e A294212    0,   1,    3,    4,    5, ...
%e A294212    0,   1,    3,    5,    6, ...
%e A294212 Square array A(n,k) begins:
%e A294212    1,   1,    1,    1,    1, ...
%e A294212    0,   1,    1,    1,    1, ...
%e A294212    0,   3,    5,    5,    5, ...
%e A294212    0,  13,   25,   31,   31, ...
%e A294212    0,  73,  193,  241,  265, ...
%e A294212    0, 501, 1601, 2261, 2501, ...
%Y A294212 Columns k=0..5 give A000007, A000262, A294213, A294214, A294215, A294216.
%Y A294212 Rows n=0 gives A000012.
%Y A294212 Main diagonal gives A058892.
%Y A294212 Cf. A058398, A294250, A294254, A294289.
%K A294212 nonn,tabl
%O A294212 0,9
%A A294212 _Seiichi Manyama_, Oct 25 2017