A106261 Expansion of 1/sqrt(1 - 20*x - 20*x^2).

1, 10, 160, 2800, 51400, 970000, 18640000, 362800000, 7128700000, 141103000000, 2809273600000, 56197096000000, 1128614356000000, 22741607080000000, 459548117440000000, 9309106936000000000, 188980474087000000000

Offset: 0

Author

Paul Barry, Apr 28 2005

Keywords

• easy
• nonn

Comments

Central coefficient of (1 + 10x + 30x^2)^n. Tenth binomial transform of 1/sqrt(1 - 120x^2). In general, 1/sqrt(1 - 4*r*x - 4*r*x^2) has e.g.f. exp(2rx)*BesselI(0,2r*sqrt((r+1)/r)x)), and a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k gives the central coefficient of (1 + (2r)*x + r(r+1)*x^2) and is the (2r)-th binomial transform of 1/sqrt(1 - 8*C(n+1,2)x^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..750
• 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. A006139, A106258, A106259, A106260.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-20*x-20*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2013 *)

PARI

for(n=0,25, print1(sum(k=0,n,binomial(2*k,k)*binomial(k,n-k)*5^k), ", ")) \\ G. C. Greubel, Jan 31 2017

Formula

E.g.f.: exp(10*x)*BesselI(0, 10*sqrt(6/5)*x).

a(n) = Sum_{k=0..n} C(2k, k)*C(k, n-k)*5^k.

D-finite with recurrence: n*a(n) + 10*(-2*n+1)*a(n-1) + 20*(-n+1)*a(n-2) = 0. - R. J. Mathar, Nov 26 2012

a(n) ~ sqrt((1+sqrt(5/6))/2) * (10+2*sqrt(30))^n / sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2013