OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
Vaclav Kotesovec, Graph - The asymptotic ratio
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
FORMULA
E.g.f.: prod{m >= 1} 1/(1-x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 14 2002
a(n) = R(n,1), R(n,m) = R(n,m+1)+binomial(n,m)*(m-1)!*R(n-m,m), R(n,n)=(n-1)!, R(n,m)=0 for n<m. - Vladimir Kruchinin, Sep 09 2014
a(n) ~ c * n! * n, where c = exp(-gamma) = 0.56145948..., where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Mar 05 2016
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018
MAPLE
p := product(1/(1-x^m/m), m=1..100):
s := series(p, x, 100):
for i from 0 to 100 do printf(`%.0f, `, i!*coeff(s, x, i)) od:
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 21 2014
MATHEMATICA
nmax = 25; CoefficientList[Series[1/Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
nmax = 25; CoefficientList[Series[Exp[Sum[PolyLog[j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
PROG
(PARI)
N=66; q='q+O('q^N);
f=1/prod(n=1, N, 1-1/n*q^n );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */
(Maxima)
R(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then (n-1)! else R(n, m+1)+binomial(n, m)*(m-1)!*R(n-m, m);
makelist(R(n, 1), n, 0, 21); /* Vladimir Kruchinin, Sep 09 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Jan 09 2001
Prepended a(0) = 1, Joerg Arndt, Oct 06 2012
STATUS
approved