A165431 A transform of the central binomial coefficients.
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
Offset: 0
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).
Crossrefs
Cf. A026569.
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(8*x^3-4*x+1))); // G. C. Greubel, Oct 20 2018 -
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
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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 *) -
PARI
x='x+O('x^30); Vec(1/sqrt(8*x^3-4*x+1)) \\ G. C. Greubel, Oct 20 2018
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
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