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A294188 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(k*(1/(1-x)^k - 1)).

%I A294188 #15 Oct 24 2017 11:35:57
%S A294188 1,1,0,1,1,0,1,4,3,0,1,9,28,13,0,1,16,117,256,73,0,1,25,336,1881,2848,
%T A294188 501,0,1,36,775,8416,35505,37024,4051,0,1,49,1548,27925,241696,763209,
%U A294188 547936,37633,0,1,64,2793,75888,1134025,7769856,18309861,9064192
%N A294188 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(k*(1/(1-x)^k - 1)).
%H A294188 Seiichi Manyama, <a href="/A294188/b294188.txt">Antidiagonals n = 0..139, flattened</a>
%F A294188 A(0,k) = 1 and A(n,k) = k^2 * (n-1)! * Sum_{j=1..n} binomial(j+k-1,k)*A(n-j,k)/(n-j)! for n > 0.
%e A294188 Square array A(n,k) begins:
%e A294188    1,   1,     1,      1,       1, ...
%e A294188    0,   1,     4,      9,      16, ...
%e A294188    0,   3,    28,    117,     336, ...
%e A294188    0,  13,   256,   1881,    8416, ...
%e A294188    0,  73,  2848,  35505,  241696, ...
%e A294188    0, 501, 37024, 763209, 7769856, ...
%Y A294188 Columns k=0..3 give A000007, A000262, A294189, A294190.
%Y A294188 Rows n=0..1 give A000012, A000290.
%Y A294188 Main diagonal gives A294192.
%Y A294188 Cf. A291709, A294046.
%K A294188 nonn,tabl
%O A294188 0,8
%A A294188 _Seiichi Manyama_, Oct 24 2017