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

0,2

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

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

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 *)

Crossrefs

Cf. A002893, A002895, A169715.

Sequence in context: A365564 A189122 A062023 * A084095 A174495 A121414

Adjacent sequences: A169711 A169712 A169713 * A169715 A169716 A169717

Keyword

nonn

Author

N. J. A. Sloane, Apr 17 2010

Status

approved