A168643 Triangle read by rows (0 <= k <= n): T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)*(x + 1)^n - (5 - n)*(x^n + 1) for 1 <= n <= 4, and p (x,n) = 4*(x + 1)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n - j))) for n >= 5.

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 8, 12, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 280, 224, 84, 16, 1, 1, 18, 108, 336, 504, 504, 336, 108, 18, 1, 1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1

Offset

0,5

Links

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

Example

Triangle begins:
  1;
  1,  1;
  1,  8,   1;
  1,  9,   9,   1;
  1,  8,  12,   8,   1;
  1, 10,  30,  30,  10,    1;
  1, 12,  45,  80,  45,   12,   1;
  1, 14,  63, 140, 140,   63,  14,   1;
  1, 16,  84, 224, 280,  224,  84,  16,   1;
  1, 18, 108, 336, 504,  504, 336, 108,  18,  1;
  1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1;
  ...

Mathematica

p[x_, n_] := If[n == 0, 1, If[n == 1, x + 1, 4*(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)]]];
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 <= 4 then (6 - n)*binomial(n, k) else ratcoef(4*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 2), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

Crossrefs

Cf. A132046, A168641, A168644, A168646.

Sequence in context: A109571 A366494 A133823 * A173742 A146881 A174301

Adjacent sequences: A168640 A168641 A168642 * A168644 A168645 A168646

Keyword

nonn,easy,tabl,less

Author

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

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

Status

approved