0,3

Row n contains 1 + n(n-1)/2 entries. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2009]

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]

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.

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]

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.

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]

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 this_sequence A162750 A075680 A160166

Adjacent sequences: A067308 A067309 A067310 this_sequence A067312 A067313 A067314

nonn

Henry Bottomley (se16(AT)btinternet.com), Jan 14 2002