cp's OEIS Frontend

A329497 Numerator of the rational number alpha_n involved in the calculation of the second moment of the n-th term of Ulam's "history-dependent random sequence".

%I A329497 #12 Dec 20 2021 14:37:16
%S A329497 0,2,21,223,2603,33623,477977,2474153,125009419,2268871883,8825650861,
%T A329497 183051309251,89588582779,8538921797353,51275872341193,
%U A329497 59590264767344263,322723582109367139,45705953681299945463,79490015605061949133
%N A329497 Numerator of the rational number alpha_n involved in the calculation of the second moment of the n-th term of Ulam's "history-dependent random sequence".
%H A329497 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 A329497 0, 2, 21/2, 223/6, 2603/24, 33623/120, 477977/720, 2474153/1680, ...
%p A329497 s:=[1]; a:=[0];
%p A329497 for N from 2 to 40 do # N = n+1
%p A329497 n:=N-1;
%p A329497 t1:=s[n]+(1/n)*add(s[k],k=1..n)+2*s[n]/n+(2/n)*a[n];
%p A329497 t2:=s[n]+(1/n)*add(s[k],k=1..n)+a[n]+(2/n)*add(a[k],k=1..n);
%p A329497 s:=[op(s),t1];
%p A329497 a:=[op(a),t2];
%p A329497 od:
%p A329497 s; # sigma_n
%p A329497 a; # alpha_n
%p A329497 sn:=map(numer,s); # A329495
%p A329497 sd:=map(denom,s); # A329496
%p A329497 an:=map(numer,a); # A329497
%p A329497 ad:=map(denom,a); # A329498
%Y A329497 Cf. A329495, A329496, A327498.
%K A329497 nonn,frac
%O A329497 1,2
%A A329497 _N. J. A. Sloane_, Nov 17 2019