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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)!).

%I A275586 #34 Feb 16 2025 08:33:36
%S A275586 1,2,5,9,14,20,27,28,35,42,44,48,54,65,75,77,90,104,110,119,132,135,
%T A275586 152,154,165,170,189,208,209,230,252,273,275,297,299,324,350,377,405,
%U A275586 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
%N 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)!).
%C A275586 Integers that do not appear uniquely in the Catalan triangle A009766.
%H A275586 Giovanni Resta, <a href="/A275586/b275586.txt">Table of n, a(n) for n = 1..10000</a>
%H A275586 D. F. Bailey, <a href="http://www.maa.org/sites/default/files/D11233._F._Bailey.pdf">Counting arrangements of 1's and -1's</a>, Mathematics Magazine, 69 (1996): 128-131.
%H A275586 Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, Tony W. H. Wong, <a href="https://www.emis.de/journals/JIS/VOL22/Fiorini/fiorini3.html">Primes and Perfect Powers in the Catalan Triangle</a>, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
%H A275586 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CatalansTriangle.html">Catalan's Triangle</a>
%e A275586 The Catalan triangle (A009766) starts:
%e A275586 1
%e A275586 1, 1
%e A275586 1, 2, 2
%e A275586 1, 3, 5,  5
%e A275586 1, 4, 9, 14, 14
%e A275586 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.
%e A275586 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
%o A275586 (Python)
%o A275586 def remove_duplicates(values):
%o A275586     output = []
%o A275586     seen = set()
%o A275586     for value in values:
%o A275586         if value not in seen:
%o A275586             output.append(value)
%o A275586             seen.add(value)
%o A275586     return output
%o A275586 def Non_Unique_Catalan_Triangle(k):
%o A275586     t = []
%o A275586     t.append([])
%o A275586     t[0].append(1)
%o A275586     for h in range(1, k):
%o A275586         t.append([])
%o A275586         t[0].append(1)
%o A275586     for i in range(1, k):
%o A275586         for j in range(0, k):
%o A275586             if i>j:
%o A275586                 t[i].append(0)
%o A275586             else:
%o A275586                 t[i].append(t[i-1][j] + t[i][j-1])
%o A275586     l = []
%o A275586     for r in range(0, k):
%o A275586         for s in range(0, k):
%o A275586             l.append(t[r][s])
%o A275586     non_unique = []
%o A275586     for n in l:
%o A275586         if  n <= k and n>1 and l.count(n) > 1:
%o A275586             non_unique.append(n)
%o A275586     non_unique = remove_duplicates(non_unique)
%o A275586     print (non_unique)
%Y A275586 Cf. A009766, A275481 (complement).
%K A275586 nonn
%O A275586 1,2
%A A275586 _Edgar Jaramillo Rodriguez_, _Nathaniel Benjamin_, _Eric Jovinelly_, _Eugene Fiorini_, Aug 02 2016