A106247 Expansion of (1+2*x-x^2-2*x^3+x^4) / (1-x^2)^3.

1, 2, 2, 4, 4, 6, 7, 8, 11, 10, 16, 12, 22, 14, 29, 16, 37, 18, 46, 20, 56, 22, 67, 24, 79, 26, 92, 28, 106, 30, 121, 32, 137, 34, 154, 36, 172, 38, 191, 40, 211, 42, 232, 44, 254, 46, 277, 48, 301, 50, 326, 52, 352, 54, 379, 56, 407, 58, 436, 60, 466, 62, 497, 64, 529, 66

Offset: 0

Author

Paul Barry, Apr 26 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A106246. Transform of C(2,n)=(1,2,1,0,0,0,...) under the mapping that takes g(x) to (1/(1-x^2))g(x/(1-x^2)).

Links

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

Programs

Mathematica

CoefficientList[Series[(1+2x-x^2-2x^3+x^4)/(1-x^2)^3,{x,0,100}],x] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{1,2,2,4,4,6},100] (* Harvey P. Dale, Mar 23 2023 *)

PARI

Vec((1+2*x-x^2-2*x^3+x^4)/(1-x^2)^3 + O(x^100)) \\ Colin Barker, Jul 23 2016

Formula

a(n)=sum{k=0..floor(n/2), C(n-k, k)C(2, n-2k)}; a(2n)=A000124(n); a(2n+1)=A005843(n+1).

From Colin Barker, Jul 23 2016: (Start)

a(n) = (16+10*n+(-1)^n*(-6+n)*n+n^2)/16.

a(n) = (n^2+2*n+8)/8 for n even.

a(n) = n+1 for n odd.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5.

(End)