OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Master's thesis, 1992. [see page 24]
N. J. A. Sloane, Transforms
FORMULA
G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A011782(k).
Euler transform of c(n) with g.f.: -x*(-2*x^2-x+2)/(-4*x^3+2*x^2+2*x-1). - Simon Plouffe, Jun 20 2018
a(n) ~ A247003 * 2^(n-1) * exp(2*sqrt(n) - 1/2 + c) / (sqrt(Pi)*n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032... - Vaclav Kotesovec, Sep 15 2021
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
2^(d-1), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
end:
a:= n-> add(g(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..35); # Alois P. Heinz, May 22 2018
# Maple program to compute c(n) from a(n) or a(n) from c(n).
with(numtheory):
andrews:=proc(liste) local n, z, serie, ls, i, d, aaa;
n:=nops(liste);
aaa:=liste;
serie:=listtoseries(aaa, z, ogf):
ls:=series(ln(serie), z, n);
[seq(coeff(ls, z, d), d=1..n)];
[seq(elemmobius(%, i), i=1..n-1)]
end:
swerdna:=proc(liste) local n, i, z;
n:=nops(liste);
series(convert([seq((1-z^i)^(-liste[i]), i=1..n)], `*`), z, n);
[seq(coeff(%, z, i), i=0..n-1)]
end:
elemmobius:=proc(liste, d) local k, rep;
rep:=0;
for k in divisors(d) do
rep:=rep+liste[k]*mobius(iquo(d, k))/iquo(d, k)
od;
rep
end:
# Here andrews() finds the c(n) and swerdna() finds the a(n) if the c(n) are known.
# For ordinary partitions the c(n) are [1, 1, 1, 1, 1, ...].
# Simon Plouffe, Jun 20 2018
MATHEMATICA
nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 22 2018
STATUS
approved