A169714 The function W_5(2n) (see Borwein et al. reference for definition).

1, 5, 45, 545, 7885, 127905, 2241225, 41467725, 798562125, 15855173825, 322466645545, 6687295253325, 140927922498025, 3010302779775725, 65046639827565525, 1419565970145097545, 31249959913055650125, 693192670456484513025

Offset: 0

Author

N. J. A. Sloane, Apr 17 2010

Keywords

• nonn

Comments

Row sums of the fourth power of A008459. - Peter Bala, Mar 05 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J. M. Borwein, A short walk can be beautiful, 2015.
• Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
• Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals (2012)
• Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012

Crossrefs

Cf. A002893, A002895, A169715.

Programs

Maple

A169714 := proc(n)
    W(5,2*n) ;
end proc: # with W() from A169715, R. J. Mathar, Mar 27 2012

Mathematica

a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^5, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 30 2013, after Peter Bala *)
max = 17; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 4] (* Jean-François Alcover, Mar 24 2015, after Peter Bala *)

Formula

Sum_{n>=0} a(n)*x^n/n!^2 = (Sum_{n>=0} x^n/n!^2)^5 = BesselI(0, 2*sqrt(x))^5. - Peter Bala, Mar 05 2013

D-finite with recurrence: n^4*a(n) = (35*n^4 - 70*n^3 + 63*n^2 - 28*n + 5)*a(n-1) - (n-1)^2*(259*n^2 - 518*n + 285)*a(n-2) + 225*(n-2)^2*(n-1)^2*a(n-3). - Vaclav Kotesovec, Mar 09 2014

a(n) ~ 5^(2*n+5/2) / (16 * Pi^2 * n^2). - Vaclav Kotesovec, Mar 09 2014