The On-Line Encyclopedia of Integer Sequences (OEIS)

A182818 revision #33

A182818

G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).

1, 3, 8, 19, 41, 83, 161, 299, 538, 942, 1610, 2694, 4427, 7153, 11387, 17884, 27741, 42543, 64565, 97034, 144519, 213432, 312720, 454803, 656835, 942364, 1343596, 1904354, 2684008, 3762667, 5248002, 7284132, 10063319, 13841107, 18956002

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OFFSET

0,2

COMMENTS

sigma(2n) = A000203(2n), the sum of divisors of 2n (A062731).

Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: E(x^2)/E(x)^3 where E(x)=prod(n>=1, 1-x^n). [Joerg Arndt, Dec 05 2010]

Conjecture: exp( sum(n>=1, sigma(s*n)*x^n/n) ) == prod( d divides s, eta(x^d)^(-moebius(d)*sigma(s/d)) ). [Joerg Arndt, Dec 05 2010]

The ordinary generating function is the infinite product A(x) * A(x^2) * A(x^3) * ..., where A(x) is the ordinary generating function of A005408. - Gary W. Adamson, Jul 15 2012

EXAMPLE

G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...

log(A(x)) = 3*x + 7*x^2/2 + 12*x^3/3 + 15*x^4/4 + 18*x^5/5 + 28*x^6/6 + 24*x^7/7 + 31*x^8/8 +...+ sigma(2n)*x^n/n +...

MATHEMATICA

nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(2*m)*x^m/m)+x*O(x^n)), n)}

(PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)^3) \\ Joerg Arndt, Dec 05 2010]

CROSSREFS

Cf. A062731, A000203, A000041; variants: A182819, A182820, A182821.

Sequence in context: A328540 A260547 A328541 * A095846 A153732 A089924

Adjacent sequences: A182815 A182816 A182817 * A182819 A182820 A182821

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 05 2010

STATUS

editing