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A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).

%I A109692 #39 Jul 14 2023 18:33:21
%S A109692 1,1,1,1,4,3,1,9,23,15,1,16,86,176,105,1,25,230,950,1689,945,1,36,505,
%T A109692 3480,12139,19524,10395,1,49,973,10045,57379,177331,264207,135135,1,
%U A109692 64,1708,24640,208054,1038016,2924172,4098240,2027025
%N A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
%C A109692 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.
%C A109692 T(n,k), 0 <= k <= n, is the number of elements in the Coxeter group B_n with absolute length k. - _Jose Bastidas_, Jul 14 2023
%H A109692 Seiichi Manyama, <a href="/A109692/b109692.txt">Rows n = 0..139, flattened</a>
%F A109692 T(n,m) = T(n-1,m) + (2*n-1)*T(n-1,m-1) with T(n,n) = (2*n-1)!! and T(n,0) = 1. - _Johannes W. Meijer_, Jun 08 2009
%e A109692 Triangle T(n,k) begins:
%e A109692   1;
%e A109692   1,  1;
%e A109692   1,  4,   3;
%e A109692   1,  9,  23,   15;
%e A109692   1, 16,  86,  176,   105;
%e A109692   1, 25, 230,  950,  1689,   945;
%e A109692   1, 36, 505, 3480, 12139, 19524, 10395;
%e A109692   ...
%p A109692 nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1,m) + (2*n-1)*a(n-1,m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # _Johannes W. Meijer_, Jun 08 2009, revised Nov 25 2012
%Y A109692 Cf. A039758 (signed version). A028338 transposed.
%Y A109692 Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341.
%Y A109692 Row sums: A000165.
%Y A109692 Central terms: A293318.
%Y A109692 Cf. A161198 (transposed scaled triangle version).
%K A109692 nonn,tabl
%O A109692 0,5
%A A109692 _Philippe Deléham_, Aug 08 2005