login
The function W_5(2n) (see Borwein et al. reference for definition).
8

%I #48 Apr 05 2024 11:10:14

%S 1,5,45,545,7885,127905,2241225,41467725,798562125,15855173825,

%T 322466645545,6687295253325,140927922498025,3010302779775725,

%U 65046639827565525,1419565970145097545,31249959913055650125,693192670456484513025

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

%C Row sums of the fourth power of A008459. - _Peter Bala_, Mar 05 2013

%H Vincenzo Librandi, <a href="/A169714/b169714.txt">Table of n, a(n) for n = 0..200</a>

%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.

%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="http://www.carmamaths.org/resources/jon/walks.pdf">Random Walk Integrals</a>, 2010.

%H Jonathan M. Borwein and Armin Straub, <a href="https://www.carmamaths.org/resources/jon/wmi-paper.pdf">Mahler measures, short walks and log-sine integrals</a> (2012)

%H Armin Straub, <a href="http://arminstraub.com/pub/dissertation">Arithmetic aspects of random walks and methods in definite integration</a>, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012

%F 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

%F 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

%F a(n) ~ 5^(2*n+5/2) / (16 * Pi^2 * n^2). - _Vaclav Kotesovec_, Mar 09 2014

%p A169714 := proc(n)

%p W(5,2*n) ;

%p end proc: # with W() from A169715, _R. J. Mathar_, Mar 27 2012

%t 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_ *)

%t 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_ *)

%Y Cf. A002893, A002895, A169715.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Apr 17 2010