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G.f.: exp( Sum_{n>=1} A056789(n)*x^n/n ), where A056789(n) = Sum_{k=1..n} lcm(k,n)/gcd(k,n).
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%I #19 Oct 28 2024 07:11:39

%S 1,1,2,5,10,23,40,86,159,300,559,1037,1887,3400,6102,10763,19027,

%T 33138,57621,99160,169934,289432,490208,826169,1385272,2312155,

%U 3840729,6354981,10467872,17179510,28081845,45740041,74234336,120074489,193582842,311102311,498434393

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

%H Vaclav Kotesovec, <a href="/A226455/b226455.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F 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

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 23*x^5 + 40*x^6 + 86*x^7 + ...

%e where

%e 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 + ...

%t 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 *)

%o (PARI) {A056789(n)=sum(k=1,n,lcm(n,k)/gcd(n,k))}

%o {a(n)=polcoeff(exp(sum(m=1,n+1,A056789(m)*x^m/m)+x*O(x^n)),n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A023896, A056789, A226106.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 07 2013