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A174527 Triangle T(n,m) = 2*A022167(n,m) - binomial(n, m), 0 <= m <= n, read by rows.

%I A174527 #9 May 10 2019 23:39:00
%S A174527 1,1,1,1,6,1,1,23,23,1,1,76,254,76,1,1,237,2410,2410,237,1,1,722,
%T A174527 22007,67740,22007,722,1,1,2179,198905,1851507,1851507,198905,2179,1,
%U A174527 1,6552,1792492,50190504,151826374,50190504,1792492,6552,1,1,19673,16139204
%N A174527 Triangle T(n,m) = 2*A022167(n,m) - binomial(n, m), 0 <= m <= n, read by rows.
%C A174527 Row sums are 1, 2, 8, 48, 408, 5296, 113200, 4105184, 255805472, 27442457664, 5089653253824, ... = 2*A006117(n)-2^n.
%e A174527 Triangle begins
%e A174527   1;
%e A174527   1,    1;
%e A174527   1,    6,       1;
%e A174527   1,   23,      23,        1;
%e A174527   1,   76,     254,       76,         1;
%e A174527   1,  237,    2410,     2410,       237,        1;
%e A174527   1,  722,   22007,    67740,     22007,      722,       1;
%e A174527   1, 2179,  198905,  1851507,   1851507,   198905,    2179,    1;
%e A174527   1, 6552, 1792492, 50190504, 151826374, 50190504, 1792492, 6552, 1;
%p A174527 A174527 := proc(n,k)
%p A174527         2*A022167(n,k)-binomial(n,k) ;
%p A174527 end proc:
%p A174527 seq(seq(A174527(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Nov 14 2011
%t A174527 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A174527 t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
%t A174527 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%Y A174527 Cf. A060187.
%K A174527 nonn,tabl
%O A174527 0,5
%A A174527 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 21 2010