A160823 A transform of the large Schroeder numbers.
1, 1, 3, 5, 13, 27, 69, 161, 415, 1033, 2701, 6983, 18521, 49041, 131723, 354493, 962381, 2620675, 7178285, 19724513, 54430023, 150641937, 418294813, 1164528399, 3250685297, 9094701729, 25501672595, 71649158709, 201687341901
Offset: 0
Examples
G.f. = 1 + x + 3*x^2 + 5*x^3 + 13*x^4 + 27*x^5 + 69*x^6 + 161*x^7 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-x^2-Sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x)))); // G. C. Greubel, Apr 30 2018 -
Mathematica
CoefficientList[Series[(1-x-x^2-Sqrt[1-2*x-5*x^2+6*x^3+x^4])/(2*x^2*(1- x)), {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2018 *) -
PARI
x='x+O('x^50); Vec((1-x-x^2-sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x))) \\ G. C. Greubel, Apr 30 2018
Formula
G.f.: 1/(1-x-2x^2/(1-x^2/(1-x-2x^2/(1-x^2/(1-x-2x^2/(1-x^2/(1-...))))))) (continued fraction);
G.f.: (1-x-x^2-sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x)).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*A006318(k).
Conjecture: (n+2)*a(n) -3*(n+1)*a(n-1) +3(2-n)*a(n-2) +(11*n-20)*a(n-3) +(11-5*n)*a(n-4) + (4-n)*a(n-5)=0. - R. J. Mathar, Nov 16 2011
a(n) ~ sqrt((-36 + 63*sqrt(2) + sqrt(8666 - 4936*sqrt(2)))/8) * ((1 + sqrt(13 + 8*sqrt(2)))/2)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 01 2018
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