A308417 Expansion of e.g.f. exp(x*(1 + x + x^2)/(1 - x^2)^2).

1, 1, 3, 25, 145, 1461, 14011, 169933, 2231265, 32572585, 528302611, 9146070561, 174016032433, 3498446485405, 75954922790475, 1737982233878101, 42327522277348801, 1084073452000879953, 29253450397798616995, 827617575903336189865, 24503022168956714812881

Offset: 0

Author

Ilya Gutkovskiy, May 25 2019

Keywords

• nonn

Crossrefs

Cf. A007434, A026741, A082579, A088009, A301876, A308418.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[x (1 + x + x^2)/(1 - x^2)^2], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^2, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Numerator[k/2] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

PARI

my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1 + x + x^2)/(1 - x^2)^2))) \\ Michel Marcus, May 26 2019

Formula

E.g.f.: exp(Sum_{k>=1} A026741(k)*x^k).

E.g.f.: Product_{k>=1} (1 + x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434).

a(0) = 1; a(n) = Sum_{k=1..n} A026741(k)*k!*binomial(n-1,k-1)*a(n-k).

a(n) ~ 2^(-1/6) * 3^(-1/3) * n^(n - 1/6) * exp((3/2)^(4/3) * n^(2/3) - n). - Vaclav Kotesovec, May 29 2019