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%I #12 Jan 03 2019 05:01:24
%S 1,1,1,1,12,1,1,15,15,1,1,16,24,16,1,1,15,30,30,15,1,1,12,30,40,30,12,
%T 1,1,14,63,105,105,63,14,1,1,16,84,224,280,224,84,16,1,1,18,108,336,
%U 630,630,336,108,18,1,1,20,135,480,1050,1512,1050,480,135,20,1
%N Triangle read by rows (0 <= k <= n): T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(x + 1)^n - (7 - n)*(x^n + 1) for 1 <= n <= 6, and p(x,n) = 6*(x + 1)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n - j))) for n >= 7.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 15, 15, 1;
%e 1, 16, 24, 16, 1;
%e 1, 15, 30, 30, 15, 1;
%e 1, 12, 30, 40, 30, 12, 1;
%e 1, 14, 63, 105, 105, 63, 14, 1;
%e 1, 16, 84, 224, 280, 224, 84, 16, 1;
%e 1, 18, 108, 336, 630, 630, 336, 108, 18, 1;
%e 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
%e ...
%t p[x_, n_] := If[n == 0, 1, If[n == 1, x + 1, 6*(x + 1)^n - (x^n + 1) - If[n > 2, (x^n + n*x^(n - 1) + n*x + 1), (x^n + 1)] - If[ n > 3, (x^n + n*x^( n - 1) + Binomial[n, n - 2]*x^(n - 2) + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)] - If[n > 4, (x^n + n*x^( n - 1) + Binomial[n, n - 2]*x^(n - 2) + Binomial[n, n - 3]*x^(n - 3) + Binomial[ n, n - 3]*x^3 + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)] - If[n > 5, (x^n + n*x^(n - 1) + Binomial[n, n - 2]*x^( n - 2) + Binomial[n, n - 3]*x^(n - 3) + Binomial[n, n - 4]*x^( n - 4) + Binomial[n, n - 4]*x^4 + Binomial[n, n - 3]*x^3 + Binomial[n, n - 2]*x^2 + n*x + 1), (x^n + 1)]]];
%t Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
%o (Maxima) T(n,k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
%o create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 02 2019 */
%Y Cf. A132046, A168641, A168643, A168644.
%K nonn,tabl,easy,less
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E Edited by _Franck Maminirina Ramaharo_, Jan 02 2019