A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.
1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 2, 1, 4, 5, 1, 1, 1, 1, 5, 6, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 8, 1, 2, 1, 4, 1, 2, 1, 8, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 10, 1, 2, 1, 2, 5, 2, 1, 1, 1, 5, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 2, 3, 4, 1, 3, 1, 2, 3, 1, 1, 6
Offset: 0
Examples
n\k| 0 1 2 3 4 5 6 ---+-------------- 0 | 1 1 | 1 1 2 | 2 1 2 3 | 3 1 1 3 4 | 4 1 2 1 4 5 | 5 1 1 1 1 5 6 | 6 1 2 3 1 1 1 ...
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.
Programs
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Mathematica
Flatten[Join[{1},Table[(n+k)/GCD[n+k,Binomial[n+k,n]],{n,1,12},{k,0,n}]]] -
PARI
A328902(n,k)=if(n,(n+k)/gcd(binomial(n+k,n),n+k),1) \\ M. F. Hasler, Nov 04 2019
Formula
T(n, k) = (n + k)/gcd(binomial(n + k, n), n + k) for n > 0.