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

0,4

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

Example

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
...

Mathematica

Flatten[Join[{1}, Table[(n+k)/GCD[n+k, Binomial[n+k, n]], {n, 1, 12}, {k, 0, n}]]]

Prog

(PARI) A328902(n, k)=if(n, (n+k)/gcd(binomial(n+k, n), n+k), 1) \\ M. F. Hasler, Nov 04 2019

Crossrefs

Cf. A000108, A028310 (1st column), A046899, A051162, A328901 (numerator).

Sequence in context: A233864 A133232 A137152 * A159335 A109004 A103823

Adjacent sequences: A328899 A328900 A328901 * A328903 A328904 A328905

Keyword

frac,nonn,tabl

Author

Stefano Spezia, Oct 30 2019

Status

approved