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A216435 Number of Dyck n-paths with equally spaced returns. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

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.

EXAMPLE

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.

MAPLE

with(numtheory):

a:= n->`if`(n=0, 1, add((binomial(2*d-2, d-1)/d)^(n/d), d=divisors(n))):

seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2012

MATHEMATICA

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

PROG

(PARI)

C(n)=binomial(2*n, n)/(n+1);

a(n)=if(n==0, 1, sumdiv(n, d, C(d-1)^(n/d) ) );

/* Joerg Arndt, Sep 30 2012 */

CROSSREFS

Cf. A000108.

Sequence in context: A161746 A045629 A034731 * A110888 A133736 A058698

Adjacent sequences:  A216432 A216433 A216434 * A216436 A216437 A216438

KEYWORD

nonn

AUTHOR

David Scambler, Sep 10 2012

STATUS

approved

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Last modified November 19 16:06 EST 2019. Contains 329320 sequences. (Running on oeis4.)