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A165431 A transform of the central binomial coefficients. 1
1, 2, 6, 16, 46, 132, 388, 1152, 3462, 10492, 32036, 98400, 303756, 941576, 2928936, 9138176, 28584006, 89609196, 281466916, 885620576, 2790812196, 8806560056, 27823745016, 88005102336, 278637450396, 883024243032, 2800748951208 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform is 2^n.

LINKS

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

R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv:1310.2449 [cs.DM], 2013 (last line of text).

FORMULA

G.f.: 1/(1-2x-2x^2/(1-x^2/(1-2x-x^2/(1-x^2/(1-2x-x^2/(1-x^2/(1-... (continued fraction);

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

From Vaclav Kotesovec, Jul 28 2016: (Start)

D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) - 4*(2*n - 3)*a(n-3).

G.f.: 1/sqrt(8*x^3-4*x+1).

a(n) ~ sqrt(1 + 2/sqrt(5)) * (1+sqrt(5))^n / sqrt(Pi*n).

(End)

a(n) = 2^n*hypergeom([1/2, 1/2-n/2, -n/2],[1, -n],-4) for n>=1. - Peter Luschny, Jul 28 2016

MAPLE

a := n -> `if`(n=0, 1, 2^n*hypergeom([1/2, 1/2-n/2, -n/2], [1, -n], -4)):

seq(simplify(a(n)), n=0..25); # Peter Luschny, Jul 28 2016

MATHEMATICA

Table[Sum[Binomial[n-k, k]*2^(n-2*k)*Binomial[2*k, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 28 2016 *)

CoefficientList[Series[1/Sqrt[8*x^3-4*x+1], {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 28 2016 *)

PROG

(PARI) x='x+O('x^30); Vec(1/sqrt(8*x^3-4*x+1)) \\ G. C. Greubel, Oct 20 2018

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(8*x^3-4*x+1))); // G. C. Greubel, Oct 20 2018

CROSSREFS

Cf. A026569.

Sequence in context: A291036 A092687 A094039 * A182267 A003291 A148442

Adjacent sequences:  A165428 A165429 A165430 * A165432 A165433 A165434

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 18 2009

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)