A168646 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.

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, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 224, 280, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Example

Triangle begins:
  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,    1;
  1, 14,  63, 105,  105,   63,   14,   1;
  1, 16,  84, 224,  280,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...

Mathematica

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)]]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]

Prog

(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))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

Crossrefs

Cf. A132046, A168641, A168643, A168644.

Sequence in context: A010206 A174096 A070636 * A051457 A174450 A166343

Adjacent sequences: A168643 A168644 A168645 * A168647 A168648 A168649

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

Status

approved