A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

1, 1, 3, 9, 20, 52, 105, 253, 536, 1142, 2421, 4999, 10278, 20686, 41512, 81984, 161029, 312681, 603070, 1153284, 2189331, 4129537, 7733317, 14399693, 26644337, 49034811, 89741600, 163411148, 296074694, 533909026, 958416113, 1712893825, 3048468607, 5403248469, 9539609984

Offset

0,3

Comments

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

Links

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

Vaclav Kotesovec, Graph - The asymptotic ratio

Formula

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, eulerphi(d^3))*x^m/m)+x*O(x^n)), n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A068963, A002618, A061255, A299069, A301986.

Sequence in context: A364914 A220770 A191527 * A026566 A147356 A147416

Adjacent sequences: A226103 A226104 A226105 * A226107 A226108 A226109

Keyword

nonn

Author

Paul D. Hanna, May 26 2013

Status

approved