A205478 G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).

1, 1, 2, 2, 4, 4, 8, 8, 14, 15, 24, 25, 43, 45, 69, 74, 113, 120, 187, 198, 291, 314, 452, 483, 720, 770, 1089, 1182, 1657, 1784, 2530, 2724, 3764, 4102, 5593, 6053, 8361, 9049, 12183, 13304, 17831, 19378, 26097, 28355, 37548, 41107, 54031, 58894, 78008, 85052

Offset

0,3

Comments

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer series.

Links

Table of n, a(n) for n=0..49.

Formula

Logarithmic derivative yields A205479.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...

Mathematica

max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n)))))), n)}

Crossrefs

Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.

Sequence in context: A206558 A145810 A172148 * A262966 A034397 A200750

Adjacent sequences: A205475 A205476 A205477 * A205479 A205480 A205481

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved