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

1, 6, 66, 996, 18306, 384156, 8848236, 218040696, 5651108226, 152254667436, 4229523740916, 120430899525096, 3499628148747756, 103446306284890536, 3102500089343886696, 94219208840385966096, 2892652835496484004226, 89662253086458906345036

Offset: 0

Author

N. J. A. Sloane, Apr 17 2010

Keywords

• nonn

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• D. Bernstein and T. Lange, Two grumpy giants and a baby, in ANTS X, Proc. Tenth Algorithmic Number Theory Symposium, 2013.
• J. M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
• Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, FPSAC 2010, San Francisco, USA.
• Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
• 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, A008459, A169714.

Programs

Maple

W := proc(n,s)
    local a,ai ;
    if s = 0 then
        return 1;
    end if;
    a := 0 ;
    for ai in combinat[partition](s/2) do
        if nops(ai) <= n then
            af := [op(ai),seq(0,i=1+nops(ai)..n)] ;
            a := a+combinat[numbperm](af)*(combinat[multinomial](s/2,op(ai)))^2 ;
        end if ;
    end do;
    a ;
end proc:
A169715 := proc(n)
    W(6,2*n) ;
end proc: # R. J. Mathar, Mar 27 2012

Mathematica

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 *)
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 *)

Formula

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

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

a(n) ~ 3^(2*n+3) * 4^(n-1) / (Pi*n)^(5/2). - Vaclav Kotesovec, Mar 09 2014