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 A169715 The function W_6(2n) (see Borwein et al. reference for definition). 8
 1, 6, 66, 996, 18306, 384156, 8848236, 218040696, 5651108226, 152254667436, 4229523740916, 120430899525096, 3499628148747756, 103446306284890536, 3102500089343886696, 94219208840385966096, 2892652835496484004226, 89662253086458906345036 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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, Random Walk Integrals, preprint, 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 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 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 *) CROSSREFS Cf. A002893, A002895, A008459, A169714. Sequence in context: A151832 A133306 A216636 * A211824 A128319 A174496 Adjacent sequences:  A169712 A169713 A169714 * A169716 A169717 A169718 KEYWORD nonn AUTHOR N. J. A. Sloane, Apr 17 2010 STATUS approved

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Last modified January 24 13:52 EST 2020. Contains 331194 sequences. (Running on oeis4.)