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A216435
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Number of Dyck n-paths with equally spaced returns.
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2
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1, 1, 2, 3, 7, 15, 48, 133, 456, 1439, 5060, 16797, 60693, 208013, 760326, 2677217, 9879513, 35357671, 131763844, 477638701, 1790943777, 6566420517, 24748372638, 91482563641, 346597488614, 1289904685149, 4905215393598, 18370277279665, 70085754999907, 263747951750361
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OFFSET
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0,3
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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with(numtheory):
a:= n->`if`(n=0, 1, add((binomial(2*d-2, d-1)/d)^(n/d), d=divisors(n))):
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MATHEMATICA
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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
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PROG
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(PARI)
C(n)=binomial(2*n, n)/(n+1);
a(n)=if(n==0, 1, sumdiv(n, d, C(d-1)^(n/d) ) );
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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