A281593 a(n) = b(n) - Sum_{j=0..n-1} b(j) with b(n) = binomial(2*n, n).

1, 1, 3, 11, 41, 153, 573, 2157, 8163, 31043, 118559, 454479, 1747771, 6740059, 26055459, 100939779, 391785129, 1523230569, 5931153429, 23126146629, 90282147849, 352846964649, 1380430179489, 5405662979649, 21186405207549, 83101804279101, 326199124351701

Offset: 0

Author

Peter Luschny, Feb 25 2017

Keywords

• nonn

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000

Crossrefs

A279561(n) = (a(n)+1)/2.

A057552(n) = (a(n+2)-1)/2.

A162551(n) = a(n+1)-a(n).

Cf. A000984, A006134, A279557.

Programs

Maple

b := n -> binomial(2*n, n): s := n -> add(b(j), j=0..n):
a := n -> b(n) - s(n-1): seq(a(n), n=0..26);
# second program:
A281593 := series(exp(2*x)*BesselI(0, 2*x) - exp(x)*int(BesselI(0, 2*x)*exp(x), x), x = 0, 27): seq(n!*coeff(A281593, x, n), n=0..26); # Mélika Tebni, Feb 27 2024

Mathematica

a[n_] = Binomial[2n,n](1+Hypergeometric2F1[1,n+1/2,n+1,4])+I/Sqrt[3];
Table[Simplify[a[n]],{n,0,17}]
CoefficientList[Series[(2x -1)/((x -1) Sqrt[(1 -4x)]), {x, 0, 26}], x] (* Robert G. Wilson v, Feb 25 2017 *)
a[0]=1; a[n_]:=a[n-1] + 2*(n-1)*CatalanNumber[n-1];Table[a[n],{n,0,26}] (* Indranil Ghosh, Mar 03 2017 *)

PARI

a(n) = binomial(2*n,n)-sum(j=0,n-1,binomial(2*j,j)); \\ Indranil Ghosh, Mar 03 2017

PARI

c(n) = binomial(2*n,n)/(n+1);
a(n) = if(n==0,1,a(n-1) + 2*(n-1)*c(n-1)); \\ Indranil Ghosh, Mar 03 2017

Python

import math
def C(n,r): return f(n)/f(r)/f(n-r)
def A281593(n):
    s=0
    for j in range(0,n):
        s+=C(2*j,j)
    return C(2*n,n)-s # Indranil Ghosh, Mar 03 2017

Sage

def A():
    a = b = c = 1
    yield 1
    while True:
        yield a
        c = (c * (4 * b - 2)) // (b + 1)
        a += 2 * b * c
        b += 1
a = A(); print([next(a) for  in (0..25)]) # _Peter Luschny, Feb 25 2017

Formula

a(n) = [x^n] (2*x-1)/(sqrt(1-4*x)*(x-1)).

a(n) = binomial(2*n,n)*(1+hypergeom([1,n+1/2],[n+1],4))+I/sqrt(3).

a(n+1) = a(n) + 2*n*Catalan(n).

a(n) ~ (4/3)*4^n/sqrt((8*n+2)*Pi/2).

D-finite with recurrence n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jul 27 2022

E.g.f.: exp(2*x)*BesselI(0,2*x) - exp(x)*integral( BesselI(0,2*x)*exp(x) ) dx. - Mélika Tebni, Feb 27 2024