A007985 Co-growth function of the group < a, b, a^{-1}, b^{-1} | b >, isomorphic to Z (counting only reduced words).

2, 2, 6, 22, 58, 146, 414, 1206, 3410, 9634, 27670, 79926, 230666, 667058, 1936174, 5632662, 16408866, 47875650, 139908774, 409404630, 1199368026, 3517351890, 10325547582, 30339113718, 89217594738, 262561101282, 773248631222, 2278737199606

Offset

1,1

Links

Table of n, a(n) for n=1..28.

Stephen P Humphries, Cogrowth of groups and the Dedekind-Frobenius group determinant, Mathematical Proceeding of the Cambridge Philosophical Society volume 121 Part 2 pages 193-217 (1997).

Formula

G.f.: (x+1) * sqrt(1-x) / sqrt((1-3*x) * (3*x^2+1)) [from Humphries]. - Sean A. Irvine, Feb 26 2018

D-finite with recurrence n*a(n) +(-3*n+1)*a(n-1) +2*(n+1)*a(n-2) -6*n*a(n-3) +3*(-n+6)*a(n-4) +9*(n-5)*a(n-5)=0. - R. J. Mathar, Aug 15 2025

Example

Let A = a^_{-1}, B = b^{-1}. a(4) counts strings on a,A,b,B of length 4 that are reduced (no aA, Aa, bB, Bb) and have the same number of A's and a's. These are:
  bbbb, BBBB, abbA, aBBA, Abba, ABBa,
  abAb, abAB, aBAb, aBAB, babA, baBA, BabA, BaBA,
  Abab, AbaB, ABab, ABaB, bAba, bABa, BAba, BABa.

Crossrefs

Sequence in context: A004077 A271217 A202743 * A097090 A085403 A112478

Adjacent sequences: A007982 A007983 A007984 * A007986 A007987 A007988

Keyword

nonn

Author

Stephen P. Humphries

Extensions

More terms from Sean A. Irvine, Feb 26 2018

Name clarified by Andrei Zabolotskii, Apr 29 2026

Status

approved