cp's OEIS Frontend

A329496 Denominator of second moment of the n-th term of Ulam's "history-dependent random sequence".

%I A329496 #12 Dec 20 2021 14:36:55
%S A329496 1,1,2,3,8,60,240,2520,40320,181440,725760,266112,95800320,17791488,
%T A329496 1585059840,130767436800,1394852659200,35568742809600,20324995891200,
%U A329496 12164510040883200,69511485947904000,5109094217170944000,32114306507931648000
%N A329496 Denominator of second moment of the n-th term of Ulam's "history-dependent random sequence".
%H A329496 Mark Kac, <a href="https://doi.org/10.1016/0196-8858(89)90014-6">A history-dependent random sequence defined by Ulam</a>, Advances in Applied Mathematics 10.3 (1989): 270-277. See page 273, sigma_k and alpha_k.
%e A329496 1, 4, 25/2, 101/3, 655/8, 11077/60, 94199/240, 2005319/2520, ...
%p A329496 s:=[1]; a:=[0];
%p A329496 for N from 2 to 40 do # N = n+1
%p A329496 n:=N-1;
%p A329496 t1:=s[n]+(1/n)*add(s[k],k=1..n)+2*s[n]/n+(2/n)*a[n];
%p A329496 t2:=s[n]+(1/n)*add(s[k],k=1..n)+a[n]+(2/n)*add(a[k],k=1..n);
%p A329496 s:=[op(s),t1];
%p A329496 a:=[op(a),t2];
%p A329496 od:
%p A329496 s; # sigma_n
%p A329496 a; # alpha_n
%p A329496 sn:=map(numer,s); # A329495
%p A329496 sd:=map(denom,s); # A329496
%p A329496 an:=map(numer,a); # A329497
%p A329496 ad:=map(denom,a); # A329498
%Y A329496 Cf. A329495, A329497, A327498.
%K A329496 nonn,frac
%O A329496 1,3
%A A329496 _N. J. A. Sloane_, Nov 17 2019