A226106 -id:A226106 - cp's OEIS frontend

A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

1, 1, 2, 8, 15, 41, 75, 179, 378, 748, 1591, 3133, 6369, 12357, 24225, 46691, 89301, 169589, 318413, 596255, 1103468, 2036880, 3725353, 6786021, 12281026, 22107132, 39604155, 70566697, 125209095, 221048851, 388705826, 680465440, 1186649341, 2061086935

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

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

Cf. A002618, A061255, A226106, A299069.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(j + 1)/j * Sum[k*EulerPhi[k] * x^(j*k), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ exp(2^(3/2) * 7^(1/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4))) * 7^(1/8) / (2^(7/4) * 5^(1/8) * Pi^(1/4) * n^(5/8)).

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).

Original entry on oeis.org

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

Paul D. Hanna, Jun 07 2013

nonn

			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 + ...

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

Cf. A023896, A056789, A226106.

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 *)

{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),", "))

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

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A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

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