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A067311
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Triangle read by rows: T(n,k) gives number of ways of arranging n chords on a circle with k simple intersections (i.e. no intersections with 3 or more chords) - positive values only.
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6
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1, 1, 2, 1, 5, 6, 3, 1, 14, 28, 28, 20, 10, 4, 1, 42, 120, 180, 195, 165, 117, 70, 35, 15, 5, 1, 132, 495, 990, 1430, 1650, 1617, 1386, 1056, 726, 451, 252, 126, 56, 21, 6, 1, 429, 2002, 5005, 9009, 13013, 16016, 17381, 16991, 15197, 12558, 9646, 6916, 4641
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n contains 1 + n(n-1)/2 entries. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]
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REFERENCES
| John Riordan, "The Distribution of Crossings of Chords Joining Pairs of 2n Points on a Circle". Mathematics of Computation, Vol. 29, (1975), 215-222.
P. Flajolet and M. Noy, Analytic combinatorics of chord diagrams; in Formal Power Series and Algebraic Combinatorics, pp. 191-201, Springer, 2000. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]
J.-G. Penaud, Une preuve bijective d'une formule de Touchard-Riordan, Discrete Math., 139, 1995, 347-360. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]
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LINKS
| alt.math.recreational discussion
H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
P. Luschny, First 10 rows of triangle (taken from Luschny link below)
P. Luschny, Variants of Variations.
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FORMULA
| Sum_{0<=j<n} (-1)^j * C((n-j)*(n-j+1)/2-1-i, n-1) * (C(2n, j)-C(2n, j-1))
Generating polynomial of row n is (1-q)^{-n}*Sum((-1)^j*q^{j(j-1)/2}*binom(2n,n+j),j=-n..n). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]
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EXAMPLE
| Rows start: 1; 1; 2,1; 5,6,3,1; 14,28,28,20,10,4,1; 42,120,180,195,165,117,70,35,15,5,1; etc., i.e. there are 5 ways of arranging 3 chords with no intersections, 6 with one, 3 with two and 1 with three.
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MAPLE
| p := proc (n) options operator, arrow: sort(simplify((sum((-1)^j*q^((1/2)*j*(j-1))*binomial(2*n, n+j), j = -n .. n))/(1-q)^n)) end proc; for n from 0 to 7 do seq(coeff(p(n), q, i), i = 0 .. (1/2)*n*(n-1)) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]
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MATHEMATICA
| nmax = 15; se[n_] := se[n] = Series[ Sum[(-1)^j*q^(j(j-1)/2)*Binomial[2 n, n+j], {j, -n, n}]/(1-q)^n , {q, 0, nmax}];
t[n_, k_] := Coefficient[se[n], q^k]; t[n_, 0] = Binomial[2 n, n]/(n + 1);
Select[Flatten[Table[t[n, k], {n, 0, nmax}, {k, 0, 2nmax}] ], Positive] [[1 ;; 55]]
(* From Jean-François Alcover, Jun 22 2011, after E. Deutsch *)
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CROSSREFS
| Row sums are A001147 (double factorials). Columns include A000108 (Catalan) for k=0 and A002694 for k=1. A067310 has a different view of the same table.
Sequence in context: A062991 A118984 A073474 * A162750 A075680 A192024
Adjacent sequences: A067308 A067309 A067310 * A067312 A067313 A067314
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KEYWORD
| nonn,tabf
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jan 14 2002
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