A119975 E.g.f. exp(x)*(Bessel_I(0,2*sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).
1, 2, 7, 22, 77, 266, 947, 3382, 12217, 44338, 161855, 593110, 2181445, 8046650, 29759147, 110303798, 409655281, 1524056546, 5678827511, 21189499030, 79164147389, 296094973418, 1108623865123, 4154794910518, 15584520425641
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
m:=50; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3/2)*((1+3*x)/(6*x*Sqrt(1-2*x-7*x^2)) -1/(6*x)))); // G. C. Greubel, Aug 17 2018 -
Mathematica
Table[Sum[Binomial[n,k]*Binomial[k,Floor[k/2]]*2^Floor[k/2],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 26 2013 *) -
PARI
x='x+O('x^50); Vec(3/2*((1+3*x)/(6*x*sqrt(1-2*x-7*x^2))-1/(6*x))) \\ G. C. Greubel, Mar 19 2017
Formula
G.f.: 3/2*((1+3*x)/(6*x*sqrt(1-2*x-7*x^2))-1/(6*x)). - corrected by Vaclav Kotesovec, Jun 26 2013
a(n) = Sum_{k=0..n} C(n,k)*C(k,floor(k/2))2^floor(k/2);
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j-k)*C(k,j-k)*2^(j-k).
a(n) = (1/pi)*int(x^n*(x+3)/(4*sqrt(-x^2+2x+7)), x, 1-2*sqrt(2), 1+2*sqrt(2)).
a(n) = (1/pi)*int((x+1)^n*(x+4)/(4*sqrt(8-x^2)),x,-2*sqrt(2),2*sqrt(2)).
Conjecture: (n+1)*a(n) +(n-4)*a(n-1) +(9-13*n)*a(n-2) +21*(2-n)*a(n-3) =0. - R. J. Mathar, Dec 10 2011
a(n) ~ sqrt(16+11*sqrt(2))*(1+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 26 2013
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