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
Examples
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 ...
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.
Crossrefs
Programs
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Mathematica
Flatten[Join[{1},Table[LCM[Binomial[n+k,n],n+k]/(n+k),{n,1,11},{k,0,n}]]]
Formula
T(n, k) = lcm(binomial(n + k, n), n + k)/(n + k) for n > 0.