cp's OEIS Frontend

A165431 A transform of the central binomial coefficients.

%I A165431 #30 Sep 08 2022 08:45:47
%S A165431 1,2,6,16,46,132,388,1152,3462,10492,32036,98400,303756,941576,
%T A165431 2928936,9138176,28584006,89609196,281466916,885620576,2790812196,
%U A165431 8806560056,27823745016,88005102336,278637450396,883024243032,2800748951208
%N A165431 A transform of the central binomial coefficients.
%C A165431 Hankel transform is 2^n.
%H A165431 G. C. Greubel, <a href="/A165431/b165431.txt">Table of n, a(n) for n = 0..1000</a>
%H A165431 R. De Castro, A. L. Ramírez and J. L. Ramírez, <a href="http://arxiv.org/abs/1310.2449">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, arXiv:1310.2449 [cs.DM], 2013 (last line of text).
%F A165431 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);
%F A165431 a(n) = Sum_{k=0..n} C(n-k,k)*2^(n-2k)*C(2k,k).
%F A165431 From _Vaclav Kotesovec_, Jul 28 2016: (Start)
%F A165431 D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) - 4*(2*n - 3)*a(n-3).
%F A165431 G.f.: 1/sqrt(8*x^3-4*x+1).
%F A165431 a(n) ~ sqrt(1 + 2/sqrt(5)) * (1+sqrt(5))^n / sqrt(Pi*n).
%F A165431 (End)
%F A165431 a(n) = 2^n*hypergeom([1/2, 1/2-n/2, -n/2],[1, -n],-4) for n>=1. - _Peter Luschny_, Jul 28 2016
%p A165431 a := n -> `if`(n=0,1,2^n*hypergeom([1/2, 1/2-n/2, -n/2],[1, -n],-4)):
%p A165431 seq(simplify(a(n)),n=0..25); # _Peter Luschny_, Jul 28 2016
%t A165431 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 *)
%t A165431 CoefficientList[Series[1/Sqrt[8*x^3-4*x+1], {x, 0, 30}], x] (* _Vaclav Kotesovec_, Jul 28 2016 *)
%o A165431 (PARI) x='x+O('x^30); Vec(1/sqrt(8*x^3-4*x+1)) \\ _G. C. Greubel_, Oct 20 2018
%o A165431 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(8*x^3-4*x+1))); // _G. C. Greubel_, Oct 20 2018
%Y A165431 Cf. A026569.
%K A165431 easy,nonn
%O A165431 0,2
%A A165431 _Paul Barry_, Sep 18 2009