A226455 G.f.: exp( Sum_{n>=1} A056789(n)*x^n/n ), where A056789(n) = Sum_{k=1..n} lcm(k,n)/gcd(k,n).

1, 1, 2, 5, 10, 23, 40, 86, 159, 300, 559, 1037, 1887, 3400, 6102, 10763, 19027, 33138, 57621, 99160, 169934, 289432, 490208, 826169, 1385272, 2312155, 3840729, 6354981, 10467872, 17179510, 28081845, 45740041, 74234336, 120074489, 193582842, 311102311, 498434393

Offset: 0

Author

Paul D. Hanna, Jun 07 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 23*x^5 + 40*x^6 + 86*x^7 + ...
where
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 51*x^5/5 + 48*x^6/6 + 148*x^7/7 + 147*x^8/8 + 253*x^9/9 + 253*x^10/10 + ... + A056789(n)*x^n/n + ...
		

Links

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

Crossrefs

Cf. A023896, A056789, A226106.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/Sqrt[1-x] * Product[1/(1 - x^k)^(k*EulerPhi[k]/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 28 2024 *)

PARI

{A056789(n)=sum(k=1,n,lcm(n,k)/gcd(n,k))}
{a(n)=polcoeff(exp(sum(m=1,n+1,A056789(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: (1/(1 - x)) * Product_{k>=2} 1/(1 - x^k)^(phi(k^2)/2), where phi() is the Euler totient function. - Ilya Gutkovskiy, May 28 2019

a(n) ~ exp(4*sqrt(Pi)*n^(3/4)/(3*5^(1/4)) + 3*zeta(3)/(2*Pi^2)) / (2^(3/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 28 2024