A146565 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,n) = (1 + x)^n for n < 2, p(x,n) = (1 + x + x^2)*(1 + x)^(n - 2) for 2 <= n <= 4, and p(x,n) = (1 + 3*x + 5*x^2 + 3*x^3 + x^4)*(1 + x)^(n - 4) for n > 4.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 8, 4, 1, 1, 5, 12, 16, 12, 5, 1, 1, 6, 17, 28, 28, 17, 6, 1, 1, 7, 23, 45, 56, 45, 23, 7, 1, 1, 8, 30, 68, 101, 101, 68, 30, 8, 1, 1, 9, 38, 98, 169, 202, 169, 98, 38, 9, 1, 1, 10, 47, 136, 267, 371, 371, 267, 136, 47, 10, 1

Offset

0,8

Comments

Row sums give A259098.

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 2, 2, 1},
  {1, 3, 4, 3, 1},
  {1, 4, 8, 8, 4, 1},
  {1, 5, 12, 16, 12, 5, 1},
  {1, 6, 17, 28, 28, 17, 6, 1},
  {1, 7, 23, 45, 56, 45, 23, 7, 1},
  {1, 8, 30, 68, 101, 101, 68, 30, 8, 1},
  {1, 9, 38, 98, 169, 202, 169, 98, 38, 9, 1},
  {1, 10, 47, 136, 267, 371, 371, 267, 136, 47, 10, 1},
  ...

Mathematica

p[x_, n_] = If[n > 0, (x + 1)^(n - 1), 0] + If[n > 2, x^2*(x + 1)^(n - 2), x^n] + If[n > 4, x^2*(x + 1)^(n - 4), 0]; Table[CoefficientList[p[x, n], x], {n, 0, 11}] // Flatten

Crossrefs

Cf. A259098.

Sequence in context: A047089 A122218 A072405 * A115594 A086623 A248736

Adjacent sequences: A146562 A146563 A146564 * A146566 A146567 A146568

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 01 2008

Extensions

Partially edited by N. J. A. Sloane, Jun 22 2015

Name clarified by Jinyuan Wang, Jul 25 2025

Status

approved