A308418 Expansion of e.g.f. exp(x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3).

1, 1, 7, 73, 649, 8821, 122311, 2064637, 37933393, 773276329, 17257075111, 414876953041, 10780187135257, 298418920103773, 8812636845668839, 275368711393020421, 9091457478119636641, 315782978460465185617, 11511089733834178827463, 439231563093877354663129

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: exp(Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k))), where J_2() is the Jordan function (A007434).

E.g.f.: Product_{k>=1} (1 + x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376).

a(n) ~ 2^(-5/4) * 21^(1/8) * n^(n - 1/8) * exp(2^(3/2) * 3^(-3/4) * 7^(1/4) * n^(3/4) - n). - Vaclav Kotesovec, May 28 2019

E.g.f.: exp(Sum_{k>=1} A308422(k)*x^k). - Ilya Gutkovskiy, May 29 2019

Mathematica

nmax = 19; CoefficientList[Series[Exp[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^3, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[1/8 (7 - (-1)^k) k^2 k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Prog

(PARI) my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1+3*x+6*x^2+3*x^3+x^4)/(1-x^2)^3))) \\ Michel Marcus, May 26 2019

Crossrefs

Cf. A007434, A059376, A088009, A255807, A308417, A308422.

Sequence in context: A117982 A003535 A050917 * A240195 A025592 A009142

Adjacent sequences: A308415 A308416 A308417 * A308419 A308420 A308421

Keyword

nonn

Author

Ilya Gutkovskiy, May 25 2019

Status

approved