A275481 Integers that appear uniquely in the Catalan triangle, A009766.

3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Author

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, and Eugene Fiorini, Jul 29 2016

Keywords

• easy
• nonn

Comments

n appears once in c_{m,k} for integers m >= k >= 1 where c_{m,k} = ((n+k)!(n-k+1))/((k)!(n+1)!).

Links

• 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, and 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

Subsequence of A007401, which is the complement of A000096.

Cf. A009766, A275586 (complement).

Programs

Mathematica

Block[{T, nn = 85}, T[n_, k_] := T[n, k] = Which[k == 0, 1, k > n, 0, True, T[n - 1, k] + T[n, k - 1]]; Rest@ Complement[Range@ nn, Union@ Flatten@ Table[T[n, k], {n, 2, nn}, {k, 2, n}]]] (* Michael De Vlieger, Feb 04 2020, after Jean-François Alcover at A009766 *)

Python

#prints the unique integers less than k
def 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])
    unique = []
    for n in l:
        if n <= k and l.count(n) == 1 :
            unique.append(n)
    print(sorted(unique))