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

%I A275481 #27 Jun 25 2025 12:06:36
%S A275481 3,4,6,7,8,10,11,12,13,15,16,17,18,19,21,22,23,24,25,26,29,30,31,32,
%T A275481 33,34,36,37,38,39,40,41,43,45,46,47,49,50,51,52,53,55,56,57,58,59,60,
%U A275481 61,62,63,64,66,67,68,69,70,71,72,73,74,76,78,79,80,81,82,83,84,85
%N A275481 Integers that appear uniquely in the Catalan triangle, A009766.
%C A275481 n appears once in c_{m,k} for integers m >= k >= 1 where c_{m,k} = ((n+k)!(n-k+1))/((k)!(n+1)!).
%H A275481 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 A275481 Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, and 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 A275481 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CatalansTriangle.html">Catalan's Triangle</a>
%t A275481 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 *)
%o A275481 (Python)
%o A275481 #prints the unique integers less than k
%o A275481 def Unique_Catalan_Triangle(k):
%o A275481     t = []
%o A275481     t.append([])
%o A275481     t[0].append(1)
%o A275481     for h in range(1, k):
%o A275481         t.append([])
%o A275481         t[0].append(1)
%o A275481     for i in range(1, k):
%o A275481         for j in range(0, k):
%o A275481             if i>j:
%o A275481                 t[i].append(0)
%o A275481             else:
%o A275481                 t[i].append(t[i-1][j] + t[i][j-1])
%o A275481     l = []
%o A275481     for r in range(0, k):
%o A275481         for s in range(0, k):
%o A275481             l.append(t[r][s])
%o A275481     unique = []
%o A275481     for n in l:
%o A275481         if n <= k and l.count(n) == 1 :
%o A275481             unique.append(n)
%o A275481     print(sorted(unique))
%Y A275481 Subsequence of A007401, which is the complement of A000096.
%Y A275481 Cf. A009766, A275586 (complement).
%K A275481 easy,nonn
%O A275481 1,1
%A A275481 _Edgar Jaramillo Rodriguez_, _Nathaniel Benjamin_, _Eric Jovinelly_, and _Eugene Fiorini_, Jul 29 2016