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A208057 Triangle by rows, generated from the odd integers and related to A000165.

%I A208057 #26 Feb 18 2025 04:03:23
%S A208057 1,1,1,4,3,1,24,18,5,1,192,144,40,7,1,1920,1440,400,70,9,1,23040,
%T A208057 17280,4800,840,108,11,1,322560,241920,67200,11760,1512,154,13,1,
%U A208057 5160960,3870720,1075200,188160,24192,2464,208,15,1
%N A208057 Triangle by rows, generated from the odd integers and related to A000165.
%C A208057 Row sums = A000165, the double factorial numbers: (1, 2, 8, 48, 384,...).
%C A208057 Left border = A002866 and the eigensequence of the odd integers prefaced with a 1.
%H A208057 Alois P. Heinz, <a href="/A208057/b208057.txt">Rows n = 0..140, flattened</a>
%F A208057 Eigentriangle of triangle A158405 (odd integers in every row: (1, 3, 5,...)); the inverse of:
%F A208057    1;
%F A208057   -1,  1;
%F A208057   -1, -3,  1;
%F A208057   -1, -3, -5,  1;
%F A208057   -1, -3, -5, -7,  1;
%F A208057   ...
%e A208057 First few rows of the triangle:
%e A208057   1;
%e A208057   1, 1;
%e A208057   4, 3, 1;
%e A208057   24, 18, 5, 1;
%e A208057   192, 144, 40, 7, 1;
%e A208057   1920, 1440, 400, 70, 9, 1;
%e A208057   23040, 17280, 4800, 840, 108, 11, 1;
%e A208057   322560, 241920, 67200, 11760, 1512, 154, 13, 1;
%e A208057   ...
%p A208057 T:= proc(n) option remember; local M;
%p A208057       M:= (Matrix(n+1, (i, j)-> `if`(i=j, 1, `if`(i>j, -2*j+1, 0)))^(-1));
%p A208057       seq(M[n+1, k], k=1..n+1)
%p A208057     end:
%p A208057 seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 27 2012
%t A208057 T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, -2*j+1, 0]], {i, 1, n+1 }, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%Y A208057 Cf. A000165, A002866, A158405.
%K A208057 nonn,tabl
%O A208057 0,4
%A A208057 _Gary W. Adamson_, Feb 22 2012
%E A208057 Typo in term 17 corrected by _Alois P. Heinz_, Dec 06 2012