A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(-1/(2*n-1)).

1, 1, 2, 8, 32, 184, 1264, 9568, 79232, 816128, 8769536, 101867776, 1322831872, 18122579968, 268425347072, 4436611211264, 73309336469504, 1303024044310528, 25235367455752192, 497968598916333568, 10431118327503650816, 234674470003955204096, 5359992446798535852032

Offset: 0

Author

Paul D. Hanna, Feb 06 2012

Keywords

• nonn

Examples

			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) * ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A000005, A001227, A028342, A168243.

Programs

Magma

m:=40;
f:= func< x | (&*[1/(1 - x^(2*n-1))^(1/(2*n-1)) : n in [1..m+10]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Dec 21 2022

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 *)

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),", "))

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. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A206303_list(m+1) # G. C. Greubel, Dec 21 2022

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