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A107415 Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).

%I A107415 #24 Sep 24 2021 16:35:03
%S A107415 1,1,-1,2,-3,1,12,-20,9,-1,288,-492,236,-33,1,34560,-59328,28812,
%T A107415 -4196,153,-1,24883200,-42750720,20803968,-3049932,114356,-873,1,
%U A107415 125411328000,-215488512000,104894749440,-15392461248,579404172,-4514276,5913,-1
%N A107415 Triangle, read by rows: T(0,0) = 1; T(n,k) = n!*T(n-1,k) - T(n-1,k-1).
%C A107415 For n>0, the row sums are 0. For n>1, sum(k=0..n) 2^k*T(n,k) = 0. The first subdiagonal (1,-3,9,-33,...) is an alternating signed version of A007489 (sum of k!, k=1..n). The first column is A000178 (superfactorials).
%C A107415 Also triangle of coefficients in expansion of Product_{k=0..n} (k! - x) in ascending powers of x. - _Seiichi Manyama_, Sep 24 2021
%H A107415 Seiichi Manyama, <a href="/A107415/b107415.txt">Rows n = 0..43, flattened</a>
%e A107415 Triangle begins
%e A107415          1;
%e A107415          1,        -1;
%e A107415          2,        -3,        1;
%e A107415         12,       -20,        9,       -1;
%e A107415        288,      -492,      236,      -33,      1;
%e A107415      34560,    -59328,    28812,    -4196,    153,   -1;
%e A107415   24883200, -42750720, 20803968, -3049932, 114356, -873, 1;
%e A107415 (1 - x) * (2 - x) = 2 - 3*x + x^2, (1 - x) * (2 - x) * (6 - x) = 12 - 20*x + 9*x^2 - x^3, etc. - _Seiichi Manyama_, Sep 24 2021
%o A107415 (PARI) t(n, k) = {if (k < 0, return (0)); if (n < k, return (0)); if (n == 0, return (1)); return (n!*t(n-1, k) - t(n-1, k-1));} \\ _Michel Marcus_, Apr 11 2013
%o A107415 (PARI) row(n) = Vecrev(prod(k=1, n, k!-x)); \\ _Seiichi Manyama_, Sep 24 2021
%Y A107415 Cf. A000178, A008955, A249677.
%K A107415 easy,sign,tabl
%O A107415 0,4
%A A107415 _Gerald McGarvey_, May 26 2005