A077938 Expansion of 1/(1-2*x-x^2-2*x^3).

1, 2, 5, 14, 37, 98, 261, 694, 1845, 4906, 13045, 34686, 92229, 245234, 652069, 1733830, 4610197, 12258362, 32594581, 86667918, 230447141, 612751362, 1629285701, 4332217046, 11519222517, 30629233482, 81442123573, 216551925662, 575804441861, 1531045056530

Offset

0,2

Comments

Number of compositions of n into two sorts of parts 1 and 3, and one sort of parts 2. - Joerg Arndt, May 02 2015

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2354

Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)

Index entries for linear recurrences with constant coefficients, signature (2,1,2)

Formula

a(n)=sum(k=1..n, sum(j=0..k, binomial(j,n-3*k+2*j)* binomial(k,j)*2^(-n+4*k-2*j))), n>0, a(0)=1. [Vladimir Kruchinin, May 05 2011]

a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3). [Stefan Schuster, Apr 24 2015]

Mathematica

LinearRecurrence[{2, 1, 2}, {1, 2, 5}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Prog

(Maxima) a(n):=sum(sum(binomial(j, n-3*k+2*j)*binomial(k, j)*2^(-n+4*k-2*j), j, 0, k), k, 1, n); // Vladimir Kruchinin, May 05 2011
(PARI) Vec(1/(1-2*x-x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Crossrefs

Absolute values of A077987.

Sequence in context: A099485 A038990 A355387 * A077987 A143141 A117294

Adjacent sequences: A077935 A077936 A077937 * A077939 A077940 A077941

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved