A109771 G.f.: sqrt(1+6*x+x^2).

1, 3, -4, 12, -44, 180, -788, 3612, -17116, 83172, -412196, 2075436, -10586892, 54595476, -284157492, 1490774076, -7875206076, 41854313412, -223636052036, 1200637707852, -6473448634348, 35037238641780, -190299310403924, 1036863750837852, -5665846701859484

Offset

0,2

Comments

G.f. = square root of weight enumerator of [4,3,2] even weight code.

a(n) gives the row sums of the coefficient array for the family Gegenbauer_C(n,-1/2,-2x-1). [From Paul Barry, Apr 20 2009]

Links

Table of n, a(n) for n=0..24.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Formula

a(n)=(-1)^n*sum{k=0..n, C(n+k-2,n-k)*C(2k,k)/(1-2k)}=(-1)^n*sum{k=0..n, C(n+k-2,n-k)*A002420(k)}; [From Paul Barry, Apr 20 2009]

G.f.: G(0)/2, where G(k)= 1 + 1/( 1 - x*(6+x)*(2*k-1)/(x*(6+x)*(2*k-1) - 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013

a(n) ~ -(-1)^n * 2^(1/4) * (1 + sqrt(2))^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 12 2017

D-finite with recurrence: n*a(n) +3*(2*n-3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Jan 25 2020

Example

1+3*x-4*x^2+12*x^3-44*x^4+180*x^5-788*x^6+3612*x^7-...

Mathematica

CoefficientList[Series[Sqrt[1+6x+x^2], {x, 0, 30}], x] (* Harvey P. Dale, Jun 30 2017 *)

Crossrefs

Sequence in context: A360992 A217477 A299809 * A052626 A336687 A298115

Adjacent sequences: A109768 A109769 A109770 * A109772 A109773 A109774

Keyword

sign

Author

N. J. A. Sloane and Nadia Heninger, Aug 13 2005

Status

approved