cp's OEIS Frontend

A329498 Denominator 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 A329498 #9 Dec 20 2021 14:37:24
%S A329498 1,1,2,6,24,120,720,1680,40320,362880,725760,7983360,2128896,
%T A329498 113218560,387459072,261534873600,836911595520,71137485619200,
%U A329498 75322043596800,1621934672117760,37429261664256000,10218188434341888000,224800145555521536000
%N A329498 Denominator 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 A329498 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 A329498 0, 2, 21/2, 223/6, 2603/24, 33623/120, 477977/720, 2474153/1680, ...
%p A329498 s:=[1]; a:=[0];
%p A329498 for N from 2 to 40 do # N = n+1
%p A329498 n:=N-1;
%p A329498 t1:=s[n]+(1/n)*add(s[k],k=1..n)+2*s[n]/n+(2/n)*a[n];
%p A329498 t2:=s[n]+(1/n)*add(s[k],k=1..n)+a[n]+(2/n)*add(a[k],k=1..n);
%p A329498 s:=[op(s),t1];
%p A329498 a:=[op(a),t2];
%p A329498 od:
%p A329498 s; # sigma_n
%p A329498 a; # alpha_n
%p A329498 sn:=map(numer,s); # A329495
%p A329498 sd:=map(denom,s); # A329496
%p A329498 an:=map(numer,a); # A329497
%p A329498 ad:=map(denom,a); # A329498
%Y A329498 Cf. A329495, A329496, A327497.
%K A329498 nonn,frac
%O A329498 1,3
%A A329498 _N. J. A. Sloane_, Nov 17 2019