OFFSET
0,2
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.
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
MATHEMATICA
CoefficientList[Series[1/Sqrt[1-20*x-20*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2013 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 28 2005
STATUS
approved