A025176 a(n) = Jacobi P-Polynomial P_n(alpha=1, beta=1, x=sqrt(2)) multiplied by 2^(n/2+floor(n/2)) and divided by n+1.

1, 2, 9, 22, 114, 308, 1717, 4902, 28526, 84284, 504410, 1525500, 9311140, 28638504, 177367949, 552532742, 3460680278, 10888711788, 68810154286, 218247846932, 1389230689596, 4435391806552, 28403131135554, 91183193910620, 586877613154252, 1892886293567192

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Jacobi polynomials

Formula

(n+2) * a(n) = 2^(2+floor(n/2)-floor((n+1)/2)) * (2n+1) * a(n-1) - 4 * (n-1) * a(n-2), with a(0)=1 and a(1)=2. - Sean A. Irvine, Aug 14 2019

From Vaclav Kotesovec, Jan 09 2023: (Start)

Recurrence: (n+1)*(n+2)*(2*n-3)*a(n) = 12*(2*n-1)*(2*n^2 - 2*n - 1)*a(n-2) - 16*(n-3)*(n-2)*(2*n+1)*a(n-4).

a(n) ~ 2^(n-1) * (1 + sqrt(2))^(n + 1/2) * (3 + 2*sqrt(2) + (-1)^n) / (sqrt(Pi) * n^(3/2)). (End)

Maple

A025176 := proc(n)
    JacobiP(n, 1, 1, sqrt(2)) ;
    %*2^(n/2+floor(n/2))/(n+1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Feb 05 2013

Mathematica

Table[ 2^(n/2+Floor[ n/2 ])/(n+1)JacobiP[ n, 1, 1, Sqrt[ 2 ] ], {n, 0, 24} ]

Crossrefs

Sequence in context: A319792 A091002 A330419 * A032315 A032224 A254710

Adjacent sequences: A025173 A025174 A025175 * A025177 A025178 A025179

Keyword

nonn

Author

Wouter Meeussen

Status

approved