A007987 Number of irreducible words of length 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero.

1, 0, 8, 40, 312, 2240, 17280, 134568, 1071000, 8627872, 70302888, 577920200, 4786740112, 39899052960, 334391846048, 2815803070920, 23809393390680, 202061204197632, 1720404406215720, 14690717541313128, 125775000062934552

Offset: 0

Author

Stephen P. Humphries

Keywords

• nonn

Comments

Also, co-growth function of a certain group given by Humphries 1997 (page 211).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Stephen P Humphries, Cogrowth of groups and the Dedekind-Frobenius group determinant, Mathematical Proc. Camb. Phil. Soc. (1997) vol. 121, pp. 193-217

Programs

Mathematica

CoefficientList[Series[(1 - x)*Hypergeometric2F1[1/12, 5/12, 1,
1728*x^4*(x - 1)*(9*x - 1)*(3*x + 1)^2/(81*x^4 - 36*x^3 - 26*x^2 - 4*x + 1)^3]/(81*x^4 - 36*x^3 - 26*x^2 - 4*x + 1)^(1/4), {x, 0,50}], x] (* G. C. Greubel, Mar 07 2017 *)

Formula

For n>0, a(n) = A168597(n) - A168597(n-1) = A002426(n)^2 - A002426(n-1)^2.

G.f.: (1-x)*hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4). - Mark van Hoeij, Apr 10 2014

Extensions

Formula and further terms from Max Alekseyev, Jun 04 2011