A098480 Expansion of 1/sqrt((1-x)^2-8x^3).

1, 1, 1, 5, 13, 25, 65, 181, 445, 1113, 2945, 7685, 19821, 51865, 136513, 358229, 942109, 2487385, 6573825, 17387045, 46066253, 122213913, 324512833, 862511605, 2294698109, 6109933657, 16280439937, 43411979845, 115835462445

Offset: 0

Author

Paul Barry, Sep 10 2004

Keywords

• easy
• nonn

Comments

1/sqrt((1-x)^2-4rx^3) expands to sum{k=0..floor(n/2), binomial(n-k,k)binomial(n-2k,k)r^k}

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2309
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Crossrefs

Cf. A098479, A098481.

Programs

Mathematica

Array[Sum[Binomial[# - k, k] Binomial[# - 2 k, k] 2^k, {k, 0, #/2}] &, 29, 0] (* Michael De Vlieger, Jul 16 2019 *)

Formula

a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(n-2k, k)2^k}. D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +(n-1)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Nov 10 2014