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%I #33 Feb 04 2020 21:41:25
%S 1,2,5,9,14,20,27,28,35,42,44,48,54,65,75,77,90,104,110,119,132,135,
%T 152,154,165,170,189,208,209,230,252,273,275,297,299,324,350,377,405,
%U 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 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 Integers that do not appear uniquely in the Catalan triangle A009766.
%H Giovanni Resta, <a href="/A275586/b275586.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/CatalansTriangle.html">Catalan's Triangle</a>
%e The Catalan triangle (A009766) starts:
%e 1
%e 1, 1
%e 1, 2, 2
%e 1, 3, 5, 5
%e 1, 4, 9, 14, 14
%e 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 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 (Python)
%o def remove_duplicates(values):
%o output = []
%o seen = set()
%o for value in values:
%o if value not in seen:
%o output.append(value)
%o seen.add(value)
%o return output
%o def Non_Unique_Catalan_Triangle(k):
%o t = []
%o t.append([])
%o t[0].append(1)
%o for h in range(1, k):
%o t.append([])
%o t[0].append(1)
%o for i in range(1, k):
%o for j in range(0, k):
%o if i>j:
%o t[i].append(0)
%o else:
%o t[i].append(t[i-1][j] + t[i][j-1])
%o l = []
%o for r in range(0, k):
%o for s in range(0, k):
%o l.append(t[r][s])
%o non_unique = []
%o for n in l:
%o if n <= k and n>1 and l.count(n) > 1:
%o non_unique.append(n)
%o non_unique = remove_duplicates(non_unique)
%o print (non_unique)
%Y Cf. A009766, A275481 (complement).
%K nonn
%O 1,2
%A _Edgar Jaramillo Rodriguez_, _Nathaniel Benjamin_, _Eric Jovinelly_, _Eugene Fiorini_, Aug 02 2016
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