OFFSET
0,2
COMMENTS
Define Q(n,x)=sum{k=0..floor(n/2), binomial(n,k)binomial(2(n-k),n)x^(n-2k)}. Then a(n)=3^n*Q(n,1/3). A084770(n) is 2^n*Q(n,1/2). Central coefficient of (1+2x+10x^2)^n.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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(2x)*BesselI(0, 2*sqrt(10)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*9^k.
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 36*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+5*sqrt(10))*(2+2*sqrt(10))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012
MATHEMATICA
Table[SeriesCoefficient[1/Sqrt[1-4*x-36*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 15 2012 *)
PROG
(PARI) x='x+O('x^66); Vec(1/sqrt(1-4*x-36*x^2)) \\ Joerg Arndt, May 11 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 08 2004
STATUS
approved