A122877 Expansion of (1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^3).

0, 1, 2, 7, 20, 65, 206, 679, 2248, 7569, 25690, 88055, 303964, 1056497, 3693158, 12977655, 45813008, 162400609, 577843890, 2063053991, 7388487460, 26535797729, 95552015614, 344897769991, 1247685613272

Offset

0,3

Comments

Binomial transform is A071357.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1723

Formula

a(n) = Sum_{k=0..n} C(n,k)*2^((k-1)/2)*C((k-1)/2+1)*(1-(-1)^k)/2, where C(n)=A000108(n).

a(n) = (1/Pi)*Integral_{x=1-2*sqrt(2)..1+2*sqrt(2)} x^n*sqrt(-x^2+2x+7)*(x-1)/8.

a(n) = (Sum_{j=0..n+1} binomial(j,n-j+3)*2^(n-j+2)*binomial(n+1,j))/(n+1). - Vladimir Kruchinin, May 19 2014

D-finite with recurrence: (n+3)*a(n) + (-3*n-4)*a(n-1) + (-5*n-1)*a(n-2) + 7*(n-2)*a(n-3) = 0. - R. J. Mathar, Feb 23 2015

a(n) ~ (1 + 2*sqrt(2))^(n + 3/2) / (sqrt(Pi) * 2^(5/4) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019

Mathematica

CoefficientList[Series[(1 - 2 x - 3 x^2 - (1 - x) Sqrt[1 - 2 x - 7 x^2])/(8 x^3), {x, 0, 24}], x] (* Michael De Vlieger, Apr 17 2020 *)

Prog

(Maxima)
a(n):=sum(binomial(j, n-j+3)*2^(n-j+2)*binomial(n+1, j), j, 0, n+1)/(n+1); /* Vladimir Kruchinin, May 19 2014 */

Crossrefs

Sequence in context: A000935 A035071 A055891 * A192680 A000150 A318232

Adjacent sequences: A122874 A122875 A122876 * A122878 A122879 A122880

Keyword

easy,nonn

Author

Paul Barry, Sep 16 2006

Status

approved