A103978 Expansion of (sqrt(1-12*x^2)+12*x^2+2*x-1)/(2*x*sqrt(1-12*x^2)).

1, 3, 6, 9, 54, 54, 540, 405, 5670, 3402, 61236, 30618, 673596, 288684, 7505784, 2814669, 84440070, 28146690, 956987460, 287096238, 10909657044, 2975361012, 124965162504, 31241290626, 1437099368796, 331638315876, 16581915793800

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1855

Formula

G.f.: 1/sqrt(1-12*x^2)+(1-sqrt(1-12*x^2))/(2*x).

a(n) = sum{k=0..floor(n/2), 3^(n-k) * A000108(k) * C(k+1, n-k)}.

D-finite with recurrence: -(n+1)*a(n)+2*(n-1)*a(n-1) +12*(2n-3)*a(n-2) +24(2-n)*a(n-3) + 144*(4-n)*a(n-4)=0. - R. J. Mathar, Dec 14 2011

a(n) ~ 2^(n + 1/2) * 3^(n/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^(n + 1/2) * 3^((n+1)/2) / (sqrt(Pi) * n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021

Maple

rec:= -(n+1)*a(n)+2*(n-1)*a(n-1)+12*(2*n-3)*a(n-2)+24*(2-n)*a(n-3)+144*(4-n)*a(n-4):
f:= gfun:-rectoproc({rec=0, a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 9}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Sep 13 2020

Mathematica

CoefficientList[Series[(Sqrt[1-12x^2]+12x^2+2x-1)/(2x Sqrt[1-12x^2]), {x, 0, 30}], x] (* Harvey P. Dale, Aug 06 2022 *)

Crossrefs

Cf. A025225, A059304, A103973.

Sequence in context: A038224 A133195 A196156 * A289064 A293537 A073910

Adjacent sequences: A103975 A103976 A103977 * A103979 A103980 A103981

Keyword

easy,nonn

Author

Paul Barry, Feb 23 2005

Status

approved