A212205 G.f.: ((1+2*x)*sqrt(1-6*x^2+x^4)-1+5*x^2-2*x^3)/(2*x*(1-6*x^2)).

1, 1, 2, 4, 8, 18, 36, 86, 172, 426, 852, 2162, 4324, 11166, 22332, 58438, 116876, 309042, 618084, 1648154, 3296308, 8851206, 17702412, 47813790, 95627580, 259585002, 519170004, 1415431266, 2830862532, 7747200558, 15494401116, 42545600310, 85091200620, 234346445154, 468692890308, 1294260644906, 2588521289812, 7165245015510, 14330490031020

Offset: 0

Author

N. J. A. Sloane, May 11 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.

Programs

Maple

a:= proc(n) option remember; `if`(n<6, [1, 1, 2, 4, 8, 18][n+1],
      (-132*a(n-1) +(660-834*n+84*n^2)*a(n-2) +804*a(n-3)
      +(2981*n-6690-259*n^2)*a(n-4) -72*a(n-5) +6*(n-6)*(7*n-59)*a(n-6))
      / ((n+1)*(7*n-66)))
    end:
seq(a(n), n=0..40); # Alois P. Heinz, May 21 2013

Mathematica

CoefficientList[Series[((1+2*x)*Sqrt[1-6*x^2+x^4]-1+5*x^2-2*x^3)/(2*x*(1-6*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, May 21 2013 *)

Formula

a(n) ~ 2*2^(3/4)*(1+sqrt(2))^(n+3)/(n^(3/2)*sqrt(Pi)) if n is even and a(n) ~ 2^(3/4)*(1+sqrt(2))^(n+4)/(n^(3/2)*sqrt(Pi)) if n is odd. - Vaclav Kotesovec, May 21 2013

Conjecture D-finite with recurrence: (n+1)*a(n) +2*(n-1)*a(n-1) +6*(-2*n+1)*a(n-2) +12*(-2*n+5)*a(n-3) +(37*n-77)*a(n-4) +2*(37*n-151)*a(n-5) +6*(-n+5)*a(n-6) +12*(-n+7)*a(n-7)=0. - R. J. Mathar, Aug 20 2018