A065490 Exponents in expansion of constant A065463 as Product_{n>1} zeta(n)^(-a(n)).

0, 1, -1, 1, -2, 3, -4, 5, -8, 13, -18, 25, -40, 62, -90, 135, -210, 324, -492, 750, -1164, 1809, -2786, 4305, -6710, 10460, -16264, 25350, -39650, 62057, -97108, 152145, -238818, 375165, -589520, 927200, -1459960, 2300346, -3626200

Offset: 1

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• sign

Comments

The sequence 1,1,1,1,2,3,4,5,8,13,18,25,40,62,90,135,... appears in Lehrer-Segal on p. 285, in the following context: Let V=Sum_{k>=1} V_k be the graded vector space H_*(PC^oo)[1], which has Poincaré series [or Poincare series] p(t)=t/(1-t^2). This sequence gives the dimensions of the free graded Lie algebra L on V.

Inverse Euler transform of F(1-n) where F() is Fibonacci numbers A000045. - Michael Somos, Jul 21 2003

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000
• G. I. Lehrer, Some sequences arising at the interface of representation theory and homotopy theory
• G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290.
• G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A065463.

Programs

Mathematica

a[n_] := DivisorSum[n, (-1)^#*MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[# -1]-1)&]/n; Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)

PARI

a(n)=if(n<1,0,sumdiv(n,d,(-1)^d*moebius(n/d)*(fibonacci(d+1)+fibonacci(d-1)-1))/n)

Formula

a(n) = (1/n)*Sum_{d|n} (-1)^d*mu(n/d)*(Fibonacci(d-1)+Fibonacci(d+1)-1). - Vladeta Jovovic, May 03 2003

a(n) ~ (-1)^n * phi^n / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2019

Extensions

More terms and formula from Christian G. Bower, Aug 23 2002