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A169715 The function W_6(2n) (see Borwein et al. reference for definition).

%I A169715 #63 Apr 05 2024 11:10:14
%S A169715 1,6,66,996,18306,384156,8848236,218040696,5651108226,152254667436,
%T A169715 4229523740916,120430899525096,3499628148747756,103446306284890536,
%U A169715 3102500089343886696,94219208840385966096,2892652835496484004226,89662253086458906345036
%N A169715 The function W_6(2n) (see Borwein et al. reference for definition).
%C A169715 Row sums of the fifth power of A008459. - _Peter Bala_, Mar 05 2013
%C A169715 a(n)/6^(2n) is the probability that two throws of n 6-sided dice will give the same result - _Henry Bottomley_, Aug 30 2016
%H A169715 Vincenzo Librandi, <a href="/A169715/b169715.txt">Table of n, a(n) for n = 0..200</a>
%H A169715 D. Bernstein and T. Lange, <a href="http://dx.doi.org/10.2140/obs.2013.1.87">Two grumpy giants and a baby</a>, in ANTS X, Proc. Tenth Algorithmic Number Theory Symposium, 2013.
%H A169715 J. M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
%H A169715 Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="https://www.carmamaths.org/resources/jon/walks.pdf">Some Arithmetic Properties of Short Random Walk Integrals</a>, FPSAC 2010, San Francisco, USA.
%H A169715 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>, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
%H A169715 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 A169715 Sum_{n>=0} a(n)*x^n/n!^2 = (Sum_{n>=0} x^n/n!^2)^6 = BesselI(0, 2*sqrt(x))^6. - _Peter Bala_, Mar 05 2013
%F A169715 Recurrence: n^5*a(n) = 2*(2*n-1)*(14*n^4 - 28*n^3 + 28*n^2 - 14*n + 3)*a(n-1) - 4*(n-1)^3*(196*n^2 - 392*n + 255)*a(n-2) + 1152*(n-2)^2*(n-1)^2*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Mar 09 2014
%F A169715 a(n) ~ 3^(2*n+3) * 4^(n-1) / (Pi*n)^(5/2). - _Vaclav Kotesovec_, Mar 09 2014
%p A169715 W := proc(n,s)
%p A169715     local a,ai ;
%p A169715     if s = 0 then
%p A169715         return 1;
%p A169715     end if;
%p A169715     a := 0 ;
%p A169715     for ai in combinat[partition](s/2) do
%p A169715         if nops(ai) <= n then
%p A169715             af := [op(ai),seq(0,i=1+nops(ai)..n)] ;
%p A169715             a := a+combinat[numbperm](af)*(combinat[multinomial](s/2,op(ai)))^2 ;
%p A169715         end if ;
%p A169715     end do;
%p A169715     a ;
%p A169715 end proc:
%p A169715 A169715 := proc(n)
%p A169715     W(6,2*n) ;
%p A169715 end proc: # _R. J. Mathar_, Mar 27 2012
%t A169715 a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^6, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Dec 30 2013, after _Peter Bala_ *)
%t A169715 max = 17; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 5] (* _Jean-François Alcover_, Mar 24 2015, after _Peter Bala_ *)
%Y A169715 Cf. A002893, A002895, A008459, A169714.
%K A169715 nonn
%O A169715 0,2
%A A169715 _N. J. A. Sloane_, Apr 17 2010