A243814 Expansion of -(x*(1-sqrt((2*(1-sqrt(4*x^2+1)))/x+1)))/(1-sqrt(4*x^2+1)) - 1.

0, 1, 2, 4, 10, 29, 86, 259, 806, 2573, 8332, 27301, 90498, 302933, 1022074, 3472148, 11868242, 40788053, 140851104, 488485195, 1700694884, 5941906864, 20826229564, 73208454375, 258031793698, 911704863957, 3228661787336

Offset

0,3

Links

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

Formula

a(n) = Sum(i=0..floor((n-1)/2), (-1)^i*binomial(n,i)*binomial(2*n-4*i,n -2*i +1))/n, n>0, a(0)=0.

G.f. A(x) satisfies A(x) = x*(A(x)^4 + 4*A(x)^3 + 5*A(x)^2 + 4*A(x) + 1) / (A(x)^2 + 2*A(x)+1).

a(n) ~ 15^(n+1/2) / (4^n * n^(3/2) * sqrt(17*Pi)). - Vaclav Kotesovec, Jun 15 2014

Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-1)*a(n) -n*(47*n^2-127*n+86)*a(n-1) +4*(38*n^3-186*n^2+291*n-145)*a(n-2) +4*(-94*n^3+724*n^2-1887*n+1650)*a(n-3) +16*(n-4)*(64*n^2-382*n+595)*a(n-4) -16*(n-5)*(47*n-174)*(n-4)*a(n-5) +1920*(n-5)*(n-6)*(n-4)*a(n-6)=0. - R. J. Mathar, Jan 25 2020

Mathematica

CoefficientList[Series[-1 + 2/(1 + Sqrt[(2 + x - 2*Sqrt[1 + 4*x^2])/x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)

Prog

(Maxima)
a(n):=if n=0 then 0 else sum((-1)^i*binomial(n, i)*binomial(2*n-4*i, n-2*i+1), i, 0, (n-1)/2)/n;
(PARI) x='x+O('x^30); concat([0], Vec(-(x*(1-sqrt((2*(1-sqrt(4*x^2+1)))/x+1)))/(1-sqrt(4*x^2+1))-1)) \\ G. C. Greubel, Oct 06 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(-(x*(1-Sqrt((2*(1-Sqrt(4*x^2+1)))/x+1)))/(1-Sqrt(4*x^2+1))-1)); // G. C. Greubel, Oct 06 2018

Crossrefs

Sequence in context: A148112 A224845 A148113 * A005505 A262592 A187255

Adjacent sequences: A243811 A243812 A243813 * A243815 A243816 A243817

Keyword

nonn

Author

Vladimir Kruchinin, Jun 11 2014

Status

approved