A270490 a(n) = Sum_{i=0..(n+1)/2} binomial(2*i+1,i)*binomial(2*n-2*i,n)/(2*i+1).

1, 2, 7, 24, 87, 320, 1195, 4504, 17102, 65304, 250501, 964480, 3724996, 14424504, 55983091, 217702880, 848042197, 3308490496, 12924954514, 50553798696, 197948515868, 775853655760, 3043672637457, 11950142769664, 46954356540812

Offset

0,2

Links

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

Formula

G.f.: 1/x*C(C(x)^2)/(C(x)*(1-x/(1-C(x))^2)), where C(x)=(1-sqrt(1-4*x))/2.

a(n) ~ 2^(2*n+1)/sqrt(Pi*n) * (1 - Gamma(3/4)/(sqrt(Pi)*n^(1/4)) + 7*sqrt(2*Pi) / (16*n^(3/4)*Gamma(3/4))). - Vaclav Kotesovec, Mar 18 2016

Conjecture: 3*n*(n-2)*(n+2)*a(n) -4*(n+1)*(8*n^2-23*n+12)*a(n-1) +16*n *(3*n-4)*(2*n-5)*a(n-2) +8*(2*n-3)*(4*n-7)*a(n-3) -64*(2*n-5)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Jun 07 2016

Mathematica

Table[Sum[Binomial[2*i+1, i]*Binomial[2*n-2*i, n]/(2*i+1), {i, 0, (n+1)/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 18 2016 *)

Prog

(Maxima) a(n):=sum(binomial(2*i+1, i)*binomial(2*n-2*i, n)/(2*i+1), i, 0, (n+1)/2);
(PARI) for(n=0, 25, print1(sum(k=0, n, binomial(2*k+1, k)*binomial(2*n-2*k, n)/(2*k+1)), ", ")) \\ G. C. Greubel, Jun 05 2017

Crossrefs

Cf. A000108, A092392.

Sequence in context: A150389 A183876 A227824 * A104625 A221454 A151293

Adjacent sequences: A270487 A270488 A270489 * A270491 A270492 A270493

Keyword

nonn

Author

Vladimir Kruchinin, Mar 18 2016

Status

approved