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A293119 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(-x^i).

%I A293119 #23 Dec 06 2019 08:37:03
%S A293119 1,1,-1,1,0,-1,1,0,-2,-1,1,0,0,-6,1,1,0,0,-6,-12,19,1,0,0,0,-24,0,151,
%T A293119 1,0,0,0,-24,-120,240,1091,1,0,0,0,0,-120,-360,2520,7841,1,0,0,0,0,
%U A293119 -120,-720,0,21840,56519,1,0,0,0,0,0,-720,-5040,20160,181440,396271
%N A293119 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(-x^i).
%H A293119 Seiichi Manyama, <a href="/A293119/b293119.txt">Antidiagonals n = 0..139, flattened</a>
%F A293119 E.g.f. of column k: exp(x^(k+1)/(x-1)).
%F A293119 A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
%e A293119 Square array begins:
%e A293119     1,   1,    1,    1, ...
%e A293119    -1,   0,    0,    0, ...
%e A293119    -1,  -2,    0,    0, ...
%e A293119    -1,  -6,   -6,    0, ...
%e A293119     1, -12,  -24,  -24, ...
%e A293119    19,   0, -120, -120, ...
%p A293119 A:= proc(n, k) option remember; `if`(n=0, 1, -add(
%p A293119       A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
%p A293119     end:
%p A293119 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 30 2017
%t A293119 A[0, _] = 1;
%t A293119 A[n_, k_] /; 0 <= k <= n := A[n, k] = -Sum[A[n-j, k] Binomial[n-1, j-1] j!, {j, k+1, n}];
%t A293119 A[_, _] = 0;
%t A293119 Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 06 2019 *)
%Y A293119 Columns k=0..2 give A293116, A293117, A293118.
%Y A293119 Rows n=0..1 give A000012, (-1)*A000007.
%Y A293119 Main diagonal gives A000007.
%Y A293119 A(n,n-1) gives (-1)*A000142(n).
%Y A293119 Cf. A293053.
%K A293119 sign,tabl,look
%O A293119 0,9
%A A293119 _Seiichi Manyama_, Sep 30 2017