A161642 Triangle (by rows): T(n,k) = A007318(n,k) / A003989(n+1,k+1).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 2, 2, 1, 1, 5, 10, 10, 5, 1, 1, 3, 15, 5, 15, 3, 1, 1, 7, 7, 35, 35, 7, 7, 1, 1, 4, 28, 28, 14, 28, 28, 4, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 5, 15, 30, 210, 42, 210, 30, 15, 5, 1

Offset

0,8

Comments

Taking each row polynomial listed on p. 12 of the Alexeev et al. link and listing the GCD of each sub-polynomial in the indeterminate q gives the left half of this entry's symmetric/palindromic triangle. E.g., for k=6, q*s^6 + (6*q + 9*q^2) s^4 + (15*q + 15*q^2) s^2 + 5 = q*s^6 + 3*(2*q + 3*q^2)*s^4 + 15*(q + q^2)*s^2 + 5 generates (1,3,15,5). See also A055151. - Tom Copeland, Jun 18 2015

Links

Table of n, a(n) for n=0..65.

N. Alexeev, J. Andersen, R. Penner, P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014 [From Tom Copeland, Jun 18 2015]

Formula

T(2n,n) = A000108(n).

T(n,k) = binomial(n,k)/A003989(n+1,k+1), 0<=k<=n. - R. J. Mathar, Sep 04 2013

For first half (k <= floor(n/2)) of each palindromic row, T(n,k) = A055151(n,k) / A258820(n,k) = A007318(n,2k) * A000108(k) / A258820(n,k) = n! / [(n-2k)! k! (k+1)! A258820(n,k)]. - Tom Copeland, Jun 18 2015

Example

The triangle T(n,k) begins:
n\k 0 1  2  3   4   5   6  7  8 9 10 ...
0:  1
1:  1 1
2:  1 1  1
3:  1 3  3  1
4:  1 2  2  2   1
5:  1 5 10 10   5   1
6:  1 3 15  5  15   3   1
7:  1 7  7 35  35   7   7  1
8:  1 4 28 28  14  28  28  4  1
9:  1 9 36 84 126 126  84 36  9 1
10: 1 5 15 30 210  42 210 30 15 5  1
... reformatted. - Wolfdieter Lang, Aug 24 2015

Mathematica

T[n_, k_] := Binomial[n, k]/GCD[n-k+1, k+1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 06 2015, after R. J. Mathar *)

Crossrefs

Cf. A000108, A003989, A055151, A258820, A007318.

Sequence in context: A255916 A367825 A107333 * A152141 A098505 A178395

Adjacent sequences: A161639 A161640 A161641 * A161643 A161644 A161645

Keyword

nonn,tabl,easy

Author

Jason Richardson-White, Jun 15 2009

Extensions

Name changed, and R. J. Mathar's formula corrected, by Wolfdieter Lang, Aug 24 2015

Status

approved