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A208058 Triangle by rows relating to the factorials, generated from A002260.

%I A208058 #17 Mar 09 2015 06:07:15
%S A208058 1,1,1,1,2,1,2,4,3,1,6,12,9,4,1,24,48,36,16,5,1,120,240,180,80,25,6,1,
%T A208058 720,1440,1080,480,150,36,7,1,5040,10080,7560,3360,1050,252,49,8,1,
%U A208058 40320,80640,60480,26880,8400,2016,392,64,9,1
%N A208058 Triangle by rows relating to the factorials, generated from A002260.
%C A208058 Row sums = A054091: (1, 2, 4, 10, 32, 130, 652,...)
%C A208058 Left border = the factorials, A000142 prefaced with a 1.
%H A208058 Alois P. Heinz, <a href="/A208058/b208058.txt">Rows n = 0..140, flattened</a>
%F A208058 Inverse of:
%F A208058 1;
%F A208058 -1, 1;
%F A208058 1, -2, 1;
%F A208058 -1, 2, -3, 1;
%F A208058 1, -2, 3, -4, 1;
%F A208058 ..., where triangle A002260 = (1; 1,2; 1,2,3;...)
%e A208058 First few rows of the triangle =
%e A208058 1;
%e A208058 1, 1;
%e A208058 1, 2, 1;
%e A208058 2, 4, 3, 1;
%e A208058 6, 12, 9, 4, 1;
%e A208058 24, 48, 36, 16, 5, 1;
%e A208058 120, 240, 180, 80, 25, 6, 1;
%e A208058 720, 1440, 1080, 480, 150, 36, 7, 1;
%e A208058 5040, 10080, 7560, 3360, 1050, 252, 49, 8, 1;
%e A208058 ...
%p A208058 T:= proc(n) option remember; local M, k;
%p A208058       M:= Matrix(n+1, (i, j)->
%p A208058                  `if`(i=j, 1, `if`(i>j, j*(-1)^(i+j), 0)))^(-1);
%p A208058       seq(M[n+1, k], k=1..n+1)
%p A208058     end:
%p A208058 seq(T(n), n=0..14);  # _Alois P. Heinz_, Feb 24 2012
%t A208058 T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, j*(-1)^(i+j), 0]], {i, 1, n+1}, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%Y A208058 Cf. A000142, A002260, A054091.
%K A208058 nonn,tabl
%O A208058 0,5
%A A208058 _Gary W. Adamson_, Feb 22 2012