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A206303 E.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(-1/(2*n-1)). 12
1, 1, 2, 8, 32, 184, 1264, 9568, 79232, 816128, 8769536, 101867776, 1322831872, 18122579968, 268425347072, 4436611211264, 73309336469504, 1303024044310528, 25235367455752192, 497968598916333568, 10431118327503650816, 234674470003955204096, 5359992446798535852032 (list; graph; refs; listen; history; text; internal format)
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, 1, n}]/n];

a[n_] := b[n]*n!;

Table[a[n], {n, 0, 25}] (* Jean-Fran├žois 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), ", "))

CROSSREFS

Cf. A000005, A001227, A028342.

Sequence in context: A141202 A081358 A294506 * A048855 A262480 A062797

Adjacent sequences:  A206300 A206301 A206302 * A206304 A206305 A206306

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 06 2012

STATUS

approved

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Last modified April 26 12:31 EDT 2019. Contains 322472 sequences. (Running on oeis4.)