OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..448
FORMULA
a(n)/n! is the Euler transform of [1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ...].
E.g.f.: A(x) = B(x) / sqrt(B(x^2)), where B(x) = e.g.f. of A028342.
E.g.f. A(x) satisfies: Product_{n>=0} A(x^(2^n))^(1/2^n) = e.g.f. of A028342.
E.g.f.: exp(Sum_{k>=1} (d(2*k) - d(k))*x^k/k), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Sep 17 2018
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 32*x^4/4! + 184*x^5/5! + ...
The e.g.f. equals the product:
A(x) = (1-x)^(-1) * (1-x^3)^(-1/3) * (1-x^5)^(-1/5) * (1-x^7)^(-1/7) * (1-x^9)^(-1/9) * (1-x^11)^(-1/11) * ...
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(d::odd, 1, 0), d=divisors(j))*b(n-j), j=1..n)/n)
end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25); # Alois P. Heinz, Jan 24 2017
MATHEMATICA
b[n_]:= b[n]= If[n==0, 1, Sum[Sum[If[OddQ[d], 1, 0], {d, Divisors[j]}]* b[n-j], {j, n}]/n];
a[n_]:= b[n]*n!;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)
PROG
(PARI) {a(n)=n!*polcoeff(prod(m=1, n, (1-x^(2*m-1)+x*O(x^n))^(-1/(2*m-1))), n)}
for(n=0, 31, print1(a(n), ", "))
(Magma)
m:=40;
f:= func< x | (&*[1/(1 - x^(2*n-1))^(1/(2*n-1)) : n in [1..m+10]]) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Dec 21 2022
(SageMath)
m=40
def f(x): return 1/product( (1 - x^(2*n-1))^(1/(2*n-1)) for n in range(1, m+11) )
def A206303_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(x) ).egf_to_ogf().list()
A206303_list(m+1) # G. C. Greubel, Dec 21 2022
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Feb 06 2012
STATUS
approved