A125695 Expansion of (sqrt(1+2x+9x^2)+x-1)/(2x).

1, 2, -2, -2, 10, -6, -42, 102, 82, -782, 814, 3854, -12454, -5014, 98694, -142218, -472158, 1932258, 19038, -14816994, 27370410, 64159962, -334154442, 121279878, 2418497010, -5523511086, -8914677362, 61259567662, -44249714438

Offset: 0

Author

Paul Barry, Nov 30 2006, Dec 10 2008

Keywords

• easy
• sign

Comments

First column of A125694.

Hankel transform is (-2)^C(n+1, 2)*A001045(n+2).

Crossrefs

Cf. A091593.

Programs

Mathematica

CoefficientList[Series[(Sqrt[1+2x+9x^2]+x-1)/(2x),{x,0,30}],x] (* Harvey P. Dale, Jul 25 2013 *)

Formula

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

D-finite with recurrence: (n+1)*a(n) +(2*n-1)*a(n-1) +9*(n-2)*a(n-2)=0. - R. J. Mathar, Dec 02 2014