A168644 - OEIS

%I #14 Jan 03 2019 05:01:37

%S 1,1,1,1,10,1,1,12,12,1,1,12,18,12,1,1,10,20,20,10,1,1,12,45,65,45,12,

%T 1,1,14,63,140,154,63,14,1,1,16,84,224,350,252,84,16,1,1,18,108,336,

%U 630,630,384,108,18,1,1,20,135,480,1050,1260,1050,555,135,20,1

%N Triangle read by rows (0 <= k <= n): T(n,k) = [x^k] p(x,n), where p(0,0) = 1, p(x,n) = (7 - n)*binomial(n, k) - (6 - n)*(x^n + 1) for 1 <= n <= 5, and p(x,n) = 5*(x + 1)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n - j))) + (1/6)*n*(n - 1)*(n - 5)*x^(n - 3) for n >= 6.

%e Triangle begins:

%e   1;

%e   1,  1;

%e   1, 10,   1;

%e   1, 12,  12,   1;

%e   1, 12,  18,  12,    1;

%e   1, 10,  20,  20,   10,    1;

%e   1, 12,  45,  65,   45,   12,    1;

%e   1, 14,  63, 140,  154,   63,   14,   1;

%e   1, 16,  84, 224,  350,  252,   84,  16,   1;

%e   1, 18, 108, 336,  630,  630,  384, 108,  18,  1;

%e   1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1;

%e   ...

%t p[x_, n_] := If[n == 0, 1, If[n == 1, x + 1, 5*(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 - 2]*x^(n - 3) + 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 <= 5 then (7 - n)*binomial(n, k) else ratcoef(5*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 3) + (1/6)*n*(n - 1)*(n - 5)*x^(n - 3), 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, A168646.

%K nonn,easy,tabl,less

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009

%E Edited by _Franck Maminirina Ramaharo_, Jan 02 2019