A256092 G.f.: (2*x)/((1-(1-8*x)^(1/4))*(1-8*x)^(3/4)).

1, 3, 19, 132, 955, 7068, 53074, 402552, 3075747, 23632620, 182384774, 1412575112, 10972566670, 85442073304, 666717107460, 5211833678832, 40805272682787, 319915465833804, 2511182586525886, 19732752016481000, 155208152657919130, 1221848390114284680

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} binomial(2*k-1,k)*2^(n-k)*binomial(2*n-k-1,n-k)).

a(n) ~ 2^(3*n-2) / (Gamma(3/4) * n^(1/4)) * (1 + Gamma(3/4) / (sqrt(Pi) * n^(1/4)) + Gamma(3/4)^2 / (Pi*sqrt(2*n))). - Vaclav Kotesovec, Mar 15 2015

a(n) = 2^(n-1)*binomial(2*n-1,n)*(1+hypergeom([1/2,-n],[1-2*n],2)) for n >= 2. - Peter Luschny, Mar 15 2015

D-finite with recurrence +n*(n-1)*(n-2)*a(n) -12*(n-1)*(n-2)*(2*n-3)*a(n-1) +12*(n-2)*(16*n^2-64*n+65)*a(n-2) -16*(2*n-5)*(4*n-9)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Jul 27 2022

Mathematica

CoefficientList[Series[(2*x)/((1-(1-8*x)^(1/4))*(1-8*x)^(3/4)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 15 2015 *)

Prog

(Maxima)
a(n):=sum(binomial(2*k-1, k)*2^(n-k)*binomial(2*n-k-1, n-k), k, 0, n);
(Sage)
def a(n):
    h = hypergeometric([1/2, -n], [1-2*n], 2).simplify() if n != 1 else 2
    return 2^(n-1)*binomial(2*n-1, n)*(1+h)
[a(n) for n in range(22)] # Peter Luschny, Mar 15 2015
(PARI) x='x+O('x^50); Vec((2*x)/((1-(1-8*x)^(1/4))*(1-8*x)^(3/4))) \\ G. C. Greubel, Jun 03 2017

Crossrefs

Sequence in context: A199484 A063395 A226459 * A370159 A369627 A074567

Adjacent sequences: A256089 A256090 A256091 * A256093 A256094 A256095

Keyword

nonn

Author

Vladimir Kruchinin, Mar 14 2015

Status

approved