A124820 Expansion of (1-x)/(1-4*x+3*x^2-x^3).

1, 3, 9, 28, 88, 277, 872, 2745, 8641, 27201, 85626, 269542, 848491, 2670964, 8407925, 26467299, 83316385, 262271568, 825604416, 2598919345, 8181135700, 25753389181, 81069068969, 255197244033, 803335158406, 2528817970494

Offset: 0

Author

Paul Barry, Nov 08 2006

Keywords

• nonn
• easy

Comments

Row sums of A124819.

Let M = a triangle with the triangular series in every column, but the leftmost column is shifted upwards one row. Then A124820 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010

Second trisection of Narayana's cows sequence A000930. - Oboifeng Dira, Aug 03 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,1).

Programs

Mathematica

CoefficientList[Series[(1 - x)/(1 - 4 x + 3 x^2 - x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{4,-3,1},{1,3,9},30] (* Harvey P. Dale, Apr 29 2016 *)
Table[Sum[Binomial[n + 2 k + 1, 3 k + 1], {k, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Aug 03 2016 *)

PARI

a(n)=([0,1,0; 0,0,1; 1,-3,4]^n*[1;3;9])[1,1] \\ Charles R Greathouse IV, Aug 03 2016

Formula

a(n) = sum( k=0..n, C(n+2k+1, 3k+1) ).

a(n) = A052529(n+1) - A052529(n), n>1. - R. J. Mathar, Dec 15 2008