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A176468 A symmetrical triangle:q=3;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 A176468 #4 Jun 02 2025 02:52:39
%S A176468 1,1,1,1,3,1,1,11,11,1,1,37,101,37,1,1,117,473,473,117,1,1,359,-467,
%T A176468 -10331,-467,359,1,1,1087,-38805,-666047,-666047,-38805,1087,1,1,3273,
%U A176468 -564647,-22609991,-71238207,-22609991,-564647,3273,1,1,9833,-6313279
%N A176468 A symmetrical triangle:q=3;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 A176468 Row sums are:
%C A176468 {1, 2, 5, 24, 177, 1182, -10545, -1407528, -117580935, -13535434518,
%C A176468 -2541110736213,...}.
%F A176468 q=3;
%F A176468 c(n,q)=Product[1 - q^i, {i, 1, n}];
%F A176468 t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1
%e A176468 {1},
%e A176468 {1, 1},
%e A176468 {1, 3, 1},
%e A176468 {1, 11, 11, 1},
%e A176468 {1, 37, 101, 37, 1},
%e A176468 {1, 117, 473, 473, 117, 1},
%e A176468 {1, 359, -467, -10331, -467, 359, 1},
%e A176468 {1, 1087, -38805, -666047, -666047, -38805, 1087, 1},
%e A176468 {1, 3273, -564647, -22609991, -71238207, -22609991, -564647, 3273, 1},
%e A176468 {1, 9833, -6313279, -656760847, -6104652967, -6104652967, -656760847, -6313279, 9833, 1},
%e A176468 {1, 29515, -63520279, -18148426855, -499870912759, -1504945075459, -499870912759, -18148426855, -63520279, 29515, 1}
%t A176468 (*A060187*);
%t A176468 p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
%t A176468 f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
%t A176468 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A176468 t[n_, m_, q_] := f[n, m] - c[n, q]/(c[m, q]*c[n - m, q]) + 1;
%t A176468 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%Y A176468 Cf. A060187
%K A176468 sign,tabl,uned
%O A176468 0,5
%A A176468 _Roger L. Bagula_, Apr 18 2010