A205476 G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d) ).

1, 1, 2, 3, 5, 8, 12, 20, 28, 45, 65, 101, 148, 221, 316, 469, 673, 969, 1420, 2025, 2892, 4100, 5905, 8314, 11860, 16645, 23399, 32838, 46071, 64274, 89761, 124977, 173231, 240492, 332978, 460015, 634271, 874464, 1200463, 1649499, 2263102, 3098661, 4239109

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..42.

Formula

Logarithmic derivative yields A205477.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+2*x)*(1+x^2)/2 + x^3*(1+3*x)*(1+x^3)/3 + x^4*(1+4*x)*(1+2*x^2)*(1+x^4)/4 + x^5*(1+5*x)*(1+x^5)/5 + x^6*(1+6*x)*(1+3*x^2)*(1+2*x^3)*(1+x^6)/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 11*x^5/5 + 12*x^6/6 + 29*x^7/7 + 15*x^8/8 + 49*x^9/9 + 43*x^10/10 + ... + A205477(n)*x^n/n + ...

Mathematica

max = 50; s = Exp[Sum[(x^n/n)*Product[1+n*x^d/d, {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+m*x^d/d+x*O(x^n)))))), n)}

Crossrefs

Cf. A205477 (log), A198304, A205478, A205480, A205482, A205484, A205486, A205488, A205490.

Sequence in context: A357519 A179018 A352476 * A170805 A013984 A107479

Adjacent sequences: A205473 A205474 A205475 * A205477 A205478 A205479

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved