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A176469 A symmetrical triangle:q=4;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

%I A176469 #4 Jun 02 2025 02:52:44
%S A176469 1,1,1,1,2,1,1,3,3,1,1,-8,-126,-8,1,1,-103,-4114,-4114,-103,1,1,-642,
%T A176469 -82549,-353256,-82549,-642,1,1,-3281,-1430195,-23948889,-23948889,
%U A176469 -1430195,-3281,1,1,-15292,-23527496,-1548356796,-6216938526
%N A176469 A symmetrical triangle:q=4;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.
%C A176469 Row sums are:
%C A176469 {1, 2, 4, 8, -140, -8432, -519636, -50764728, -9360737692, -3387701237632,
%C A176469 -246332974040099,...}.
%F A176469 q=4;
%F A176469 c(n,q)=Product[1 - q^i, {i, 1, n}];
%F A176469 t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1
%e A176469 {1},
%e A176469 {1, 1},
%e A176469 {1, 2, 1},
%e A176469 {1, 3, 3, 1},
%e A176469 {1, -8, -126, -8, 1},
%e A176469 {1, -103, -4114, -4114, -103, 1},
%e A176469 {1, -642, -82549, -353256, -82549, -642, 1},
%e A176469 { 1, -3281, -1430195, -23948889, -23948889, -1430195, -3281, 1},
%e A176469 {1, -15292, -23527496, -1548356796, -6216938526, -1548356796, -23527496, -15292, 1},
%e A176469 {1, -67707, -380011248, -99256044576, -1594214495286, -1594214495286, -99256044576, -380011248, -67707, 1},
%e A176469 {1, -290486, -6099252663, -6353979629820, -408235051426002, -1634139479203056, -408235051426002, -6353979629820, -6099252663, -290486, 1}
%t A176469 (*A060187*);
%t A176469 p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
%t A176469 f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
%t A176469 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A176469 t[n_, m_, q_] := f[n, m] - c[n, q]/(c[m, q]*c[n - m, q]) + 1;
%t A176469 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%Y A176469 Cf. A060187
%K A176469 sign,tabl,uned
%O A176469 0,5
%A A176469 _Roger L. Bagula_, Apr 18 2010