A070190 Expansion of e.g.f. I_0(2*x)^5 + 2*Sum_{k>=1} I_k(2*x)^5, where I_n(z) are modified Bessel functions of order n.

1, 0, 10, 0, 270, 240, 10900, 25200, 551950, 2116800, 32458860, 169092000, 2120787900, 13427013600, 149506414200, 1075081207200, 11143223412750, 87198375264000, 865743970019500, 7171730187336000, 69416724049550020

Offset: 0

Author

Karol A. Penson, Apr 26 2002

Keywords

• nonn

Comments

A modification of e.g.f. of A002898, where the exponent of I, which is 3, is here replaced by 5.

U_10(n), in Labelle-Lacasse paper, number of closed paths of length n whose steps are 10th roots of unity.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Crossrefs

Cf. A002898.

Programs

Mathematica

With[{nmax = 25}, CoefficientList[Series[BesselI[0, 2*x]^5 + 2*Sum[BesselI[k, 2*x]^5, {k, 1, 2*nmax}], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 05 2018 *)

PARI

seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^5)))} \\ Andrew Howroyd, Nov 01 2018

Formula

a(n) ~ 5^(3/2) * 10^n / (4*Pi^2*n^2). - Vaclav Kotesovec, Jun 08 2021