OFFSET
0,5
COMMENTS
The g.f. z*(1-z**2-z**3-z**4+z**5)/(1-z-2*z**2+3*z**5) conjectured by Simon Plouffe in his 1992 dissertation is wrong.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..3770 (terms 0..1000 from Vincenzo Librandi)
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
FORMULA
G.f.: A(x) = x*G(x)/(x-G(x)), where G(x) = G000081(x^2), G000081(x) = x+x^2+2*x^3+ ... being the g.f. for A000081.
a(n) ~ c * d^n, where d = 1.8332964415228533737988849634129366404833316666328290543862325494628120733... is the root of the equation Sum_{k>=1} A000081(k) / d^(2*k-1) = 1 and c = 0.1345213691789841963849894554233223547113840356469443704501548999022472... - Vaclav Kotesovec, Dec 13 2020
MAPLE
G := subs(x=x^2, G000081); x*G/(x-G);
# second Maple program:
b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n, k) option remember; add(b(n+1-j*k), j=1..iquo(n, k)) end: B:= proc(n) option remember; unapply(add(b(k)*x^k, k=1..n), x) end: a:= n-> coeff(series(x* B(floor(n/2))(x^2)/ (x-B(floor(n/2))(x^2)), x=0, n+2), x, n): seq(a(n), n=0..38); # Alois P. Heinz, Aug 21 2008
MATHEMATICA
max = 38; a81[n_] := a81[n] = If[n <= 1, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j], {j, 1, n-1}]/(n-1)]; G81[x_] = Sum[a81[k]*x^k, {k, 0, max}]; G[x_] = G81[x^2]; A[x_] = x*(G[x]/(x-G[x])); CoefficientList[Series[A[x], {x, 0, max}], x] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Entry revised Mar 25 2004
STATUS
approved