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A294118 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+x)^k - 1)).

%I A294118 #18 Oct 24 2017 11:35:24
%S A294118 1,1,0,1,1,0,1,4,1,0,1,9,20,1,0,1,16,99,112,1,0,1,25,304,1233,688,1,0,
%T A294118 1,36,725,6496,16929,4544,1,0,1,49,1476,23425,152416,251829,31936,1,0,
%U A294118 1,64,2695,66816,826225,3867136,4012011,236800,1,0,1,81,4544,162337
%N A294118 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+x)^k - 1)).
%H A294118 Seiichi Manyama, <a href="/A294118/b294118.txt">Antidiagonals n = 0..139, flattened</a>
%F A294118 A(0,k) = 1 and A(n,k) = k^2 * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1)*A(n-j,k)/(n-j)! for n > 0.
%e A294118 Square array A(n,k) begins:
%e A294118    1, 1,    1,      1,       1, ...
%e A294118    0, 1,    4,      9,      16, ...
%e A294118    0, 1,   20,     99,     304, ...
%e A294118    0, 1,  112,   1233,    6496, ...
%e A294118    0, 1,  688,  16929,  152416, ...
%e A294118    0, 1, 4544, 251829, 3867136, ...
%Y A294118 Columns k=0..3 give A000007, A000012, A294119, A294120.
%Y A294118 Rows n=0..1 give A000012, A000290.
%Y A294118 Main diagonal gives A294191.
%Y A294118 Cf. A293991, A294042.
%K A294118 nonn,tabl
%O A294118 0,8
%A A294118 _Seiichi Manyama_, Oct 23 2017