A077843 Expansion of (1-x)/(1-2*x-2*x^2-2*x^3).

1, 1, 4, 12, 34, 100, 292, 852, 2488, 7264, 21208, 61920, 180784, 527824, 1541056, 4499328, 13136416, 38353600, 111978688, 326937408, 954539392, 2786910976, 8136775552, 23756451840, 69360276736, 202507008256, 591247473664, 1726229517312, 5039967998464

Offset

0,3

Comments

Invert transform of the sequence 1,3,5,5,5,5,... which has g.f. (1+2x+2x^2)/(1-x). - Paul Barry, Mar 01 2011

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2149

Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.

Index entries for linear recurrences with constant coefficients, signature (2,2,2).

Formula

a(n) = sum(k=1..n, sum(i=k..n,(sum(j=0..k, binomial(j,-3*k+2*j+i)*2^(-2*k+j+i)* binomial(k,j)))*binomial(n+k-i-1,k-1))). - Vladimir Kruchinin, May 05 2011

Example

Eigensequence of the triangle
  1,
  3, 1,
  5, 3, 1,
  5, 5, 3, 1,
  5, 5, 5, 3, 1,
  5, 5, 5, 5, 3, 1,
  5, 5, 5, 5, 5, 3, 1,
  5, 5, 5, 5, 5, 5, 3, 1,
  ...
- Paul Barry, Mar 01 2011

Mathematica

CoefficientList[Series[(1 - x)/(1 - 2*x - 2*x^2 - 2*x^3), {x, 0, 28}], x] (* Michael De Vlieger, May 02 2025 *)

Prog

(SageMath)
from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0, 1, 1, 2, 2, 2)
[next(it) for i in range(35)] # Zerinvary Lajos, Jun 25 2008
(Maxima)
a(n):=sum(sum((sum(binomial(j, -3*k+2*j+i)*2^(-2*k+j+i)*binomial(k, j), j, 0, k))*binomial(n+k-i-1, k-1), i, k, n), k, 1, n); /* Vladimir Kruchinin, May 05 2011 */
(PARI) Vec((1-x)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Crossrefs

Sequence in context: A014143 A361476 A077994 * A061703 A126948 A131593

Adjacent sequences: A077840 A077841 A077842 * A077844 A077845 A077846

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved