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A281593 a(n) = b(n) - Sum_{j=0..n-1} b(n) with b(n) = binomial(2*n, n). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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).

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);

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 *)

PROG

(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 [a.next() for _ in (0..25)]

(PARI) a(n) = binomial(2*n, n)-sum(j=0, n-1, binomial(2*j, j)); /* or */

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

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.

Sequence in context: A032952 A001835 A079935 * A113437 A076540 A196472

Adjacent sequences:  A281590 A281591 A281592 * A281594 A281595 A281596

KEYWORD

nonn

AUTHOR

Peter Luschny, Feb 25 2017

STATUS

approved

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Last modified February 20 15:04 EST 2019. Contains 320327 sequences. (Running on oeis4.)