A119976 E.g.f. exp(2x)*(Bessel_I(0,2*sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

1, 3, 12, 50, 216, 952, 4256, 19224, 87520, 400928, 1845888, 8533824, 39590656, 184216320, 859354112, 4017738112, 18820855296, 88317817344, 415075665920, 1953473141760, 9205135036416, 43425512132608, 205072796270592

Offset

0,2

Comments

Binomial transform of A119975. Binomial transform is A047781(n+1).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

G.f.: (1+2*x)/(4*x*sqrt(1-4*x-4*x^2))-1/(4*x);

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

D-finite with recurrence: (n+1)*a(n) -2*(n+2)*a(n-1) +12*(1-n)*a(n-2) +8*(2-n)*a(n-3) = 0. - R. J. Mathar, Dec 10 2011

Shorter recurrence: n*(n+1)*a(n) = 2*n*(2*n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ sqrt(20+14*sqrt(2))*(2+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

Mathematica

CoefficientList[Series[(1+2*x)/(4*x*Sqrt[1-4*x-4*x^2])-1/(4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

Prog

(PARI) x='x+O('x^50); Vec((1+2*x)/(4*x*sqrt(1-4*x-4*x^2))-1/(4*x)) \\ G. C. Greubel, Feb 08 2017
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+2*x)/(4*x*Sqrt(1-4*x-4*x^2)) -1/(4*x))); // G. C. Greubel, Aug 17 2018

Crossrefs

Sequence in context: A356280 A108080 A113441 * A074547 A151178 A151179

Adjacent sequences: A119973 A119974 A119975 * A119977 A119978 A119979

Keyword

easy,nonn

Author

Paul Barry, Jun 02 2006

Status

approved