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A174526 Triangle t(n,m) = 2*A022166(n,m)-binomial(n,m), read by rows, 0<=m<=n.

%I A174526 #15 Apr 09 2016 12:21:15
%S A174526 1,1,1,1,4,1,1,11,11,1,1,26,64,26,1,1,57,300,300,57,1,1,120,1287,2770,
%T A174526 1287,120,1,1,247,5313,23587,23587,5313,247,1,1,502,21562,194254,
%U A174526 401504,194254,21562,502,1,1,1013,86834,1575986,6619368,6619368,1575986,86834
%N A174526 Triangle t(n,m) = 2*A022166(n,m)-binomial(n,m), read by rows, 0<=m<=n.
%C A174526 Row sums are 1, 2, 6, 24, 118, 716, 5586, 58296, 834142, 16566404, 459510186,... = 2*A006116(n)-2^n.
%C A174526 The column m=1 might be A000295. - _R. J. Mathar_, Nov 14 2011
%e A174526 1;
%e A174526 1, 1;
%e A174526 1, 4, 1;
%e A174526 1, 11, 11, 1;
%e A174526 1, 26, 64, 26, 1;
%e A174526 1, 57, 300, 300, 57, 1;
%e A174526 1, 120, 1287, 2770, 1287, 120, 1;
%e A174526 1, 247, 5313, 23587, 23587, 5313, 247, 1;
%e A174526 1, 502, 21562, 194254, 401504, 194254, 21562, 502, 1;
%e A174526 1, 1013, 86834, 1575986, 6619368, 6619368, 1575986, 86834, 1013, 1;
%e A174526 1, 2036, 348457, 12695310, 107487764, 218443050, 107487764, 12695310, 348457, 2036, 1;
%p A174526 A174526 := proc(n,k)
%p A174526    2*A022166(n,k)-binomial(n,k) ;
%p A174526 end proc:
%p A174526 seq(seq(A174526(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Nov 14 2011
%t A174526 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A174526 t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
%t A174526 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%t A174526 (* second program: *)
%t A174526 t[n_, m_] := 2 QBinomial[n, m, 2] - Binomial[n, m]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 09 2016 *)
%Y A174526 Cf. A008292.
%K A174526 nonn,tabl
%O A174526 0,5
%A A174526 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 21 2010