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

1,5

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

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

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 *)

Prog

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

Crossrefs

Cf. A065463.

Sequence in context: A050024 A182153 A230771 * A214452 A282500 A222106

Adjacent sequences: A065487 A065488 A065489 * A065491 A065492 A065493

Keyword

sign

Author

N. J. A. Sloane, Nov 19 2001

Extensions

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

Status

approved