A243642 Expansion of -(4*x)/((1-sqrt(1-4*x))/2-sqrt(-3*(1-sqrt(1-4*x))+(1-sqrt(1-4*x))^2/4+1)+1).

-1, 2, 3, 10, 43, 210, 1106, 6120, 35059, 206050, 1235130, 7520860, 46385822, 289160620, 1818956184, 11531388580, 73599630147, 472547337650, 3049962382850, 19777692906300, 128789274380746, 841837751136060, 5521644155232972, 36330127521846080, 239722100410099118

Offset

0,2

Links

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

Formula

a(n) = (Sum_{k=0..(n-1)} binomial(3*n-3,n-4*k)*binomial(n+k-2,k))/(n-1), n>1, a(0) = -1, a(1)=2.

A(x) = -1/C(x)*S(x*C(x)), where C(x) is g.f. A000108, S(x) is g.f. A001003.

a(n) ~ sqrt((27-19*sqrt(2))/7) * (7+5*sqrt(2))^n / (sqrt(Pi) * n^(3/2) * 2^(n-3/4)). - Vaclav Kotesovec, Jun 15 2014

Conjecture D-finite with recurrence: +16*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(43*n-54)*a(n-1) +2*(n-2)*(52*n^2+695*n-2439)*a(n-2) +(3479*n^3-42714*n^2+169789*n-219234)*a(n-3) +2*(-4633*n^3+64020*n^2-293825*n+447594)*a(n-4) +4*(2*n-11)*(296*n^2-3643*n+11187)*a(n-5) +24*(n-7)*(2*n-11)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jan 25 2020

Mathematica

CoefficientList[Series[8*x/(-3+Sqrt[1-4*x] + Sqrt[-6+10*Sqrt[1-4*x] - 4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)

Prog

(Maxima)
a(n):=if n=0 then -1 else if n=1 then 2 else sum(binomial(3*n-3, n-4*k)*binomial(n+k-2, k), k, 0, n-1)/(n-1);
(PARI) my(x='x+O('x^50)); Vec(-(4*x)/((1-sqrt(1-4*x))/2-sqrt(-3*(1-sqrt(1-4*x))+(1-sqrt(1-4*x))^2/4+1)+1)) \\ G. C. Greubel, Jun 02 2017

Crossrefs

Cf. A000108, A001003.

Sequence in context: A037390 A024484 A007226 * A032204 A106643 A020116

Adjacent sequences: A243639 A243640 A243641 * A243643 A243644 A243645

Keyword

sign

Author

Vladimir Kruchinin, Jun 08 2014

Status

approved