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, 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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 17 2010
STATUS
approved