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%I #24 Jan 10 2015 10:48:31
%S 1,1,2,3,7,15,48,133,456,1439,5060,16797,60693,208013,760326,2677217,
%T 9879513,35357671,131763844,477638701,1790943777,6566420517,
%U 24748372638,91482563641,346597488614,1289904685149,4905215393598,18370277279665,70085754999907,263747951750361
%N Number of Dyck n-paths with equally spaced returns.
%H Vincenzo Librandi, <a href="/A216435/b216435.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0)=1, a(n) = Sum_{d|n} (binomial(2*d-2, d-1)/d)^(n/d) = Sum_{d|n} A000108(d-1)^(n/d) for n>=1.
%e The 3 Dyck 3-paths are UUUDDD*, UUDUDD* and UD*UD*UD* where * marks the returns; the paths UD*UUDD* and UUDD*UD* do not have equally spaced returns.
%p with(numtheory):
%p a:= n->`if`(n=0, 1, add((binomial(2*d-2, d-1)/d)^(n/d), d=divisors(n))):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 10 2012
%t a={1};For[n=1,n<=29,++n, t=0; d=Divisors[n];For[i=1, i<=Length[d],++i, t+= (Binomial[2*d[[i]]-2,d[[i]]-1]/d[[i]])^(n/d[[i]])];a=Append[a,t];];a
%o (PARI)
%o C(n)=binomial(2*n,n)/(n+1);
%o a(n)=if(n==0, 1, sumdiv(n,d, C(d-1)^(n/d) ) );
%o /* _Joerg Arndt_, Sep 30 2012 */
%Y Cf. A000108.
%K nonn
%O 0,3
%A _David Scambler_, Sep 10 2012
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