A308381 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(2 + x^(k^2))/(2*k^2)).

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 119296, 1075456, 9088256, 85292032, 894690304, 8968964096, 90882789376, 2409397682176, 40515889528832, 1051789297844224, 16251803853193216, 302342408330018816, 4444559976664662016, 84010278329827459072, 1289319649553742823424

Offset: 0

Author

Ilya Gutkovskiy, May 23 2019

Keywords

• nonn

Crossrefs

Cf. A008836, A053866, A205801, A306831.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (2 + x^(k^2))/(2 k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 - x^(2 k - 1))^(LiouvilleLambda[2 k - 1]/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} A053866(k)*x^k/k).

E.g.f.: Product_{k>=1} 1/(1 - x^(2*k-1))^(lambda(2*k-1)/(2*k-1)), where lambda() is the Liouville function (A008836).