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

0,3

Comments

Hankel transform is A060656(n+1).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Example

G.f. = 1 + x + 3*x^2 + 5*x^3 + 13*x^4 + 27*x^5 + 69*x^6 + 161*x^7 + ...

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 *)

Prog

(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
(Magma) m:=30; R<x>:=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

Crossrefs

Sequence in context: A026569 A035082 A005198 * A077443 A147196 A110225

Adjacent sequences: A160820 A160821 A160822 * A160824 A160825 A160826

Keyword

easy,nonn

Author

Paul Barry, May 27 2009

Status

approved