A328902 -id:A328902 - cp's OEIS frontend

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

Stefano Spezia, Oct 30 2019

			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
...

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.

Main diagonal gives A201058 (for n>0).

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

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

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

A363582 Number of admissible mesa sets among Stirling permutations of order n.

Original entry on oeis.org

1, 2, 3, 6, 12, 22, 44, 88, 169, 338, 676, 1322, 2644, 5288, 10433, 20866, 41732, 82736, 165472, 330944, 658012, 1316024, 2632048, 5242778, 10485556, 20971112, 41822049, 83644098, 167288196, 333885702, 667771404, 1335542808, 2667053601, 5334107202, 10668214404

Offset: 1

Bridget Tenner, Jun 10 2023

nonn

			For n = 4, the a(4) = 6 admissible pinnacle sets for Stirling permutations of order 4 are {}, {2}, {3}, {4}, {2,4}, and {3,4}.

Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, "Mesas of Stirling permutations," preprint.

Alois P. Heinz, Table of n, a(n) for n = 1..3323

Cf. A289871, A359066, A359067, A328901, A328902.

a:= proc(n) option remember; `if`(n<4, n, (2*n*(2*n-3)*
      a(n-1)+27*(n-4)*(n-2)*(a(n-3)/2-a(n-4)))/(n*(2*n-3)))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Jun 13 2023

Let n = 3*k+r, where r is in {0,1,2}, and let C_(x,y) be the rational Catalan numbers (A328901/A328902). Then a(n) = 2^(n-1) - Sum_{i=0..k-1} 2^(3*i+r)*C_(2*(k-i)-1,k-i).

More terms from Alois P. Heinz, Jun 13 2023

cp's OEIS Frontend

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.

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