|
%I #13 Nov 08 2017 10:48:40
%S 1,2,3,6,9,16,24,42,64,105,159,258,390,614,925,1441,2162,3317,4951,
%T 7526,11191,16841,24923,37253,54912,81493,119629,176549,258205,379025,
%U 552280,807014,1171959,1705148,2468113,3577332,5162240,7455485,10727083,15442040,22157247,31798821,45507039,65124514,92967787,132690935
%N Number of isomorphism classes of reduced Witt rings of fields with 2n orderings.
%C The number with 2n+1 orderings is the same as the number with 2n orderings (cf. A213332).
%H N. J. A. Sloane, <a href="/A213331/b213331.txt">Table of n, a(n) for n = 1..400</a>
%H Thomas C. Craven, <a href="https://doi.org/10.1016/0097-3165(80)90006-0">An application of Pólya's theory of counting to an enumeration problem arising in quadratic form theory</a>, J. Combin. Theory Ser. A 29 (1980), no. 2, 174--181. MR0583956 (81j:10027).
%p read transforms;
%p w:=proc(n) option remember; global did; local v; # did(n,d)=1 if d|n otherwise 0
%p if n=1 then 1 elif (n mod 2) = 1 then w(n-1);
%p else v:=n/2;
%p (1/n)* ( add(2*i*w(i)*did(v,i), i=1..v) +
%p add( add(2*i*w(i)*w(n-2*k)*did(k,i), i=1..k), k=1..v-1));
%p fi; end;
%p [seq(w(2*n),n=1..50)];
%Y Cf. A213332.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jun 08 2012
|
