A190667 Expansion of (1+2*x)/(1-x^4-2*x^3-2*x^2-x).

1, 3, 5, 13, 30, 69, 160, 371, 859, 1990, 4610, 10679, 24738, 57306, 132750, 307517, 712367, 1650207, 3822725, 8855390, 20513621, 47520058, 110080805, 255003553, 590718900, 1368407674, 3169933385, 7343190086, 17010591104, 39405245720

Offset: 0

Author

Dmitry Kruchinin, May 16 2011

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,2,1).

Programs

Magma

[ (n+1)*&+[ (Binomial(k, n-k+1)*Fibonacci(k))/k: k in [1..n+1] ]: n in [0..35] ]; // Klaus Brockhaus, May 17 2011

Magma

[ n eq 1 select 1 else n eq 2 select 3 else n eq 3 select 5 else n eq 4 select 13 else Self(n-1)+2*Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..36] ]; // Klaus Brockhaus, Jun 01 2011

Mathematica

CoefficientList[Series[(1+2x)/(1-x^4-2x^3-2x^2-x),{x,0,40}],x] (* or *) LinearRecurrence[ {1,2,2,1},{1,3,5,13},40] (* Harvey P. Dale, Feb 25 2023 *)

Maxima

a(n):=(n+1)*sum(binomial(k,n-k+1)*fib(k)/k,k,1,n+1);makelist(a(n),n,0,35);

Maxima

a(n):=if n<0 then 0 else if n=0 then 1 else if n=1 then 3 else if n=2 then 5 else if n=3 then 13 else a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4);

Formula

a(n) = (n+1)*sum(k=1..n+1, binomial(k, n-k+1)*A000045(k)/k).

a(n) = a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4), a(0)=1, a(1)=3, a(2)=5, a(3)=13.