A268042 - OEIS

A268042

a(n) = Ann(2n) where Ann(n) is the number of annular non-crossing matchings with n total endpoints.

1, 3, 8, 20, 57, 166, 538, 1762, 6045, 21040, 74628, 267598, 970134, 3544416, 13043650, 48283236, 179665425, 671564330, 2520312810, 9492124534, 35863942748, 135893383596, 516258841134, 1965906973886, 7502329984510, 28687263026656, 109893836400756, 421684916479018

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..27.

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

FORMULA

a(n) = Sum_{i+j=n} A268041(i, j).

PROG

(PARI) tnnk(n, k) = if (!n && !k, 1, sumdiv(gcd(n, k), d, eulerphi(d)*binomial((2*n+k)/d, n/d))/(2*n+k));

tnmk(n, m, k) = if (k==0, tnnk(n, 0)*tnnk(m, 0), k*sumdiv(gcd(k, gcd(n, m)), d, eulerphi(d)*binomial((2*n+k)/d, n/d)*binomial((2*m+k)/d, m/d))/((2*n+k)*(2*m+k)));

ann(n, m) = {if ((n+m) % 2, return (0)); if (n<m, return (ann(m, n))); sum(k=0, m, if (!((n-k)%2) && !((m-k)%2), tnmk((n-k)/2, (m-k)/2, k), 0)); }

a(n) = sum(k=0, n, ann(n-k, k));

CROSSREFS

Cf. A267998, A268041.

Sequence in context: A271843 A122228 A018790 * A009436 A354519 A302674

Adjacent sequences: A268039 A268040 A268041 * A268043 A268044 A268045

KEYWORD

nonn

AUTHOR

Michel Marcus, Jan 25 2016

STATUS

approved