cp's OEIS Frontend

A241810 Number of balanced orbitals over n sectors.

%I A241810 #16 May 20 2025 21:08:31
%S A241810 1,1,0,0,2,6,0,6,8,36,0,88,58,376,0,1096,526,4476,0,14200,5448,57284,
%T A241810 0,190206,61108,764812,0,2615268,723354,10499504,0,36677626,8908546,
%U A241810 147110276,0,522288944,113093022
%N A241810 Number of balanced orbitals over n sectors.
%C A241810 For the combinatorial definitions see A232500. An orbital is balanced if its integral is 0. The integral of an orbital w over n sectors is Sum_{k=1..n} Sum_{i=1..k} w(i) where w(i) are the jumps of the orbital represented by -1, 0, 1.
%F A241810 a(2*n) = A204459(2, n).
%F A241810 a(2*n+1) = A242087(n).
%F A241810 a(4*n) = A063074(n) = A029895(2*n) = A067059(2*n, 2*n).
%F A241810 a(4*n+2) = 0 for all n (proved by H. Havermann).
%t A241810 np[z_]:=Module[{i,j},For[i=Length[z],i>1&&z[[i-1]]>=z[[i]],i--];For[j=Length[z],z[[j]]<=z[[i-1]],j--];Join[Take[z,i-2],{z[[j]]},Reverse[Drop[ReplacePart[z,z[[i-1]],j],i-1]]]];o=Table[1,{16}];
%t A241810 n=0;f=0;Print[1];Print[1];While[n<16,n++;f=1-f;If[OddQ[f*n],Print[0],p=Join[-Take[o,n],{f},Take[o,n-f]];c=0;Do[If[Accumulate[Accumulate[p]][[-1]]==0,c++];p=np[p],{(2*n+1-f)!/(2*n!^2)}];Print[2*c]];n=n-f]
%t A241810 (* _Hans Havermann_, May 10 2014 *)
%o A241810 (Sage)
%o A241810 def A241810(n):
%o A241810     if n == 0: return 1
%o A241810     A = 0
%o A241810     T = [0] if is_odd(n) else []
%o A241810     for i in (1..n//2):
%o A241810         T.append(-1); T.append(1)
%o A241810     for p in Permutations(T):
%o A241810         P = 0; S = 0
%o A241810         for k in (0..n-1):
%o A241810             P += p[k]; S += P
%o A241810         if S == 0: A += 1
%o A241810     return A
%o A241810 [A241810(n) for n in (0..32)]
%Y A241810 Cf. A232500, A242087.
%K A241810 nonn,more
%O A241810 0,5
%A A241810 _Peter Luschny_, Apr 29 2014
%E A241810 More terms from _Hans Havermann_, May 10 2014
%E A241810 a(35), a(36) from _Hans Havermann_, May 23 2014