A256093 G.f.: 2 - x*2/(1 - (1-8*x)^(1/4)).

1, 3, 5, 20, 101, 572, 3470, 22040, 144669, 973356, 6676186, 46503080, 328034226, 2338460056, 16819478972, 121903180848, 889396747869, 6526715628492, 48141140144546, 356708675726088, 2653863473928870, 19816831149068360, 148466651633265540, 1115659552758534480

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(2*k-2,k)*2^(n-k+1)*binomial(2*n-k,n-k+1))/n, a(0)=1.

a(n) ~ 2^(3*n-4) / (Gamma(3/4) * n^(5/4)) * (1 + 2*Gamma(3/4) / (n^(1/4) * sqrt(Pi)) + 3*Gamma(3/4)^2 / (Pi*sqrt(2*n))). - Vaclav Kotesovec, Mar 15 2015

(512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3) = 0 for n >= 1. - Robert Israel, Jan 20 2020

Maple

f:= gfun:-rectoproc({(512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3), a(0) = 1, a(1) = 3, a(2) = 5, a(3) = 20}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jan 20 2020

Mathematica

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

Prog

(Maxima)
a(n):=if n=0 then 1 else sum(binomial(2*k-2, k)*2^(n-k+1)*binomial(2*n-k, n-k+1), k, 0, n+1)/n;
(PARI) x='x+O('x^50); Vec(2-x*2/(1-(1-8*x)^(1/4))) \\ G. C. Greubel, Jun 03 2017

Crossrefs

Sequence in context: A133102 A197156 A171864 * A066902 A007363 A103991

Adjacent sequences: A256090 A256091 A256092 * A256094 A256095 A256096

Keyword

nonn

Author

Vladimir Kruchinin, Mar 14 2015

Status

approved