A205479 L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n).

1, 3, 1, 7, 1, 15, 1, 15, 10, 13, 1, 67, 1, 17, 16, 63, 1, 168, 1, 37, 22, 25, 1, 555, 1, 29, 118, 49, 1, 520, 1, 543, 34, 37, 1, 1048, 1, 41, 40, 1125, 1, 554, 1, 73, 475, 49, 1, 6651, 1, 563, 52, 85, 1, 1680, 1, 1457, 58, 61, 1, 20632, 1, 65, 787, 5087, 1

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..2201

Formula

Forms the logarithmic derivative of A205478.

Example

L.g.f.: L(x) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 +...
By definition:
L(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 +...
Exponentiation yields the g.f. of A205478:
exp(L(x)) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 +...

Mathematica

max = 70; s = Sum[(x^(n-1)/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}] + O[x]^max; CoefficientList[s, x]*Range[max] (* Jean-François Alcover, Dec 23 2015 *)

Prog

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

Crossrefs

Cf. A205478 (exp), A205477, A205481, A205483, A205485, A205487, A205489, A205491.

Sequence in context: A321452 A089741 A089736 * A094024 A297172 A329034

Adjacent sequences: A205476 A205477 A205478 * A205480 A205481 A205482

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved