A328901 Triangle T(n, k) read by rows: T(n, k) is the numerator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for n > 0 and T(0, 0) = 1.

1, 1, 1, 1, 1, 3, 1, 1, 2, 10, 1, 1, 5, 5, 35, 1, 1, 3, 7, 14, 126, 1, 1, 7, 28, 21, 42, 77, 1, 1, 4, 12, 30, 66, 132, 1716, 1, 1, 9, 15, 165, 99, 429, 429, 6435, 1, 1, 5, 55, 55, 143, 1001, 715, 1430, 24310, 1, 1, 11, 22, 143, 1001, 1001, 1144, 2431, 4862, 46189, 1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 352716

Offset

0,6

Links

Stefano Spezia, First 141 rows of the triangle, flattened

D. Armstrong, N. A. Loehr, G. S. Warrington, Rational Parking Functions and Catalan Numbers, Annals of Combinatorics (2016), Volume 20, Issue 1, pp 21-58.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.

Formula

T(n, k) = lcm(binomial(n + k, n), n + k)/(n + k) for n > 0.

Example

n\k|   0   1   2   3   4   5   6
---+----------------------------
0  |   1
1  |   1   1
2  |   1   1   3
3  |   1   1   2  10
4  |   1   1   5   5  35
5  |   1   1   3   7  14 126
6  |   1   1   7  28  21  42  77
...

Mathematica

Flatten[Join[{1}, Table[LCM[Binomial[n+k, n], n+k]/(n+k), {n, 1, 11}, {k, 0, n}]]]

Crossrefs

Main diagonal gives A201058 (for n>0).

Cf. A000108, A046899, A051162, A328902 (denominator).

Sequence in context: A329473 A200702 A331599 * A362019 A016564 A104443

Adjacent sequences: A328898 A328899 A328900 * A328902 A328903 A328904

Keyword

frac,nonn,tabl

Author

Stefano Spezia, Oct 30 2019

Status

approved