A108488 Expansion of 1/sqrt(1 -2*x -3*x^2 -4*x^3 +4*x^4).

1, 1, 3, 9, 23, 69, 203, 601, 1815, 5493, 16731, 51225, 157367, 485093, 1499499, 4646233, 14427095, 44880981, 139849979, 436419737, 1363713015, 4266417221, 13362194571, 41891406681, 131452430999, 412835452213, 1297543367835

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Comments

In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has the expansion 1/sqrt(1 -2*b*x -(2*a*b -b^2)*x^2 -2*a*b^2*x^3 +(a*b)^2*x^4).

Diagonal of the rational function 1 / ((1 - x)*(1 - y) - 2*(x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A051286, A108480.

Programs

Mathematica

Table[Sum[Binomial[n-k,k]^2*2^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 24 2013 *)
CoefficientList[Series[1/Sqrt[1-2x-3x^2-4x^3+4x^4],{x,0,30}],x] (* Harvey P. Dale, Apr 06 2023 *)

PARI

{a(n)=polcoeff( exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k) * 2^k * x^k) *x^m/m) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014

Formula

a(n) = Sum_{k=0..n} C(n-k, k)^2*2^k.

a(n) ~ ((4*sqrt(2)-1)/62)^(1/4) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^(n+1) /(sqrt(Pi*n)*2^(n+2)). - Vaclav Kotesovec, Jul 24 2013

D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Aug 06 2013

G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * 2^k * x^k ). - Paul D. Hanna, Aug 31 2014