A156303 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^2)*x^n/n ), a power series in x with integer coefficients.

1, 1, 4, 8, 20, 38, 88, 162, 336, 624, 1211, 2195, 4109, 7295, 13190, 23072, 40618, 69838, 120486, 204006, 345595, 577387, 962961, 1588483, 2613930, 4262138, 6928799, 11179251, 17976330, 28720552, 45729595, 72401921, 114239033

Offset: 0

Author

Paul D. Hanna, Feb 08 2009

Keywords

• nonn

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 20*x^4 + 38*x^5 + 88*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 31*x^4/4 + 31*x^5/5 + 127*x^6/6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A001615, A301594.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*Sum[MoebiusMu[d]^2 / d, {d, Divisors @ k}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

PARI

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

PARI

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k^2)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k^2) * a(n-k) for n>0, with a(0)=1.

Euler transform of Dedekind psi function, cf. A001615. - Vladeta Jovovic, Feb 12 2009

a(n) ~ exp(3^(4/3) * (5*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) - Pi^(2/3) * n^(1/3) / (2^(4/3) * (15*Zeta(3))^(1/3)) + 1/12 - Pi^2 / (720 * Zeta(3))) * (5*Zeta(3))^(7/36) / (A * 2^(2/9) * 3^(11/36) * Pi^(29/36) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 24 2018