STATUS
reviewed
approved
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Revision History for A268041
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Table array: T(n,m) is the number of non-crossings matchings of curves embedded within an annulus with n exterior endpoints and m interior endpoints.
(history; published version)
(history; published version)
STATUS
proposed
reviewed
STATUS
editing
proposed
NAME
allocated for Michel MarcusTable array: T(n,m) is the number of non-crossings matchings of curves embedded within an annulus with n exterior endpoints and m interior endpoints.
DATA
1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 2, 1, 2, 0, 0, 0, 0, 0, 0, 4, 2, 3, 2, 3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 7, 3, 7, 3, 7, 5, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 14, 17, 8, 14, 8, 14, 8, 17, 14, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 80, 42, 48, 24, 38, 20, 34, 20, 38, 24, 48, 42, 80
OFFSET
0,11
LINKS
Paul Drube and Puttipong Pongtanapaisan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Drube/drube3.html">Annular Non-Crossing Matchings</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
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)));
a(n, m) = {if ((n+m) % 2, return (0)); if (n<m, return (a(m, n))); sum(k=0, m, if (!((n-k)%2) && !((m-k)%2), tnmk((n-k)/2, (m-k)/2, k), 0)); }
CROSSREFS
Cf. A267998.
KEYWORD
allocated
nonn,tabl
AUTHOR
Michel Marcus, Jan 25 2016
STATUS
approved
editing
NAME
allocated for Michel Marcus
KEYWORD
allocated
STATUS
approved