A275586 Numbers k that appear more than once in c_{m,n} for integers m >= n >= 1 where c_{m,n} = ((m+n)!(m-n+1))/((n)!(m+1)!).

1, 2, 5, 9, 14, 20, 27, 28, 35, 42, 44, 48, 54, 65, 75, 77, 90, 104, 110, 119, 132, 135, 152, 154, 165, 170, 189, 208, 209, 230, 252, 273, 275, 297, 299, 324, 350, 377, 405, 429, 434, 440, 464, 495, 527, 544, 560, 572, 594, 629, 637, 663, 665, 702, 740, 779, 798, 819, 860, 902, 910, 945, 950, 989

Offset: 1

Author

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, Eugene Fiorini, Aug 02 2016

Keywords

• nonn

Comments

Integers that do not appear uniquely in the Catalan triangle A009766.

Examples

			The Catalan triangle (A009766) starts:
1
1, 1
1, 2, 2
1, 3, 5,  5
1, 4, 9, 14, 14
Each entry is the sum of elements in the previous row except for those which are further right. The columns are nondecreasing, and all positive integers appear in the second column.
Since 2 appears twice in the triangle, it is in the sequence. Since 6 appears only once in the triangle, it is not in the sequence. - _Michael B. Porter_, Aug 05 2016
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• D. F. Bailey, Counting arrangements of 1's and -1's, Mathematics Magazine, 69 (1996): 128-131.
• Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, Tony W. H. Wong, Primes and Perfect Powers in the Catalan Triangle, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
• Eric W. Weisstein, Catalan's Triangle

Crossrefs

Cf. A009766, A275481 (complement).

Programs

Python

def remove_duplicates(values):
    output = []
    seen = set()
    for value in values:
        if value not in seen:
            output.append(value)
            seen.add(value)
    return output
def Non_Unique_Catalan_Triangle(k):
    t = []
    t.append([])
    t[0].append(1)
    for h in range(1, k):
        t.append([])
        t[0].append(1)
    for i in range(1, k):
        for j in range(0, k):
            if i>j:
                t[i].append(0)
            else:
                t[i].append(t[i-1][j] + t[i][j-1])
    l = []
    for r in range(0, k):
        for s in range(0, k):
            l.append(t[r][s])
    non_unique = []
    for n in l:
        if  n <= k and n>1 and l.count(n) > 1:
            non_unique.append(n)
    non_unique = remove_duplicates(non_unique)
    print (non_unique)