A329495 - OEIS

%I #8 Dec 20 2021 14:36:42

%S 1,4,25,101,655,11077,94199,2005319,62541113,530615357,3888615949,

%T 2547768661,1603125774433,510235838011,76538488318091,

%U 10462945272539099,182249306891497151,7488308007918913129,6810487755999748187

%N Numerator of second moment of the n-th term of Ulam's "history-dependent random sequence".

%H 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 1, 4, 25/2, 101/3, 655/8, 11077/60, 94199/240, 2005319/2520, ...

%p s:=[1]; a:=[0];

%p for N from 2 to 40 do # N = n+1

%p n:=N-1;

%p t1:=s[n]+(1/n)*add(s[k],k=1..n)+2*s[n]/n+(2/n)*a[n];

%p t2:=s[n]+(1/n)*add(s[k],k=1..n)+a[n]+(2/n)*add(a[k],k=1..n);

%p s:=[op(s),t1];

%p a:=[op(a),t2];

%p od:

%p s; # sigma_n

%p a; # alpha_n

%p sn:=map(numer,s); # A329495

%p sd:=map(denom,s); # A329496

%p an:=map(numer,a); # A329497

%p ad:=map(denom,a); # A329498

%Y Cf. A329496, A329497, A327498.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_, Nov 17 2019