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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)!).
3
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
1,2
COMMENTS
Integers that do not appear uniquely in the Catalan triangle A009766.
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, 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
EXAMPLE
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
PROG
(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)
CROSSREFS
Cf. A009766, A275481 (complement).
Sequence in context: A266450 A361742 A132296 * A333449 A075543 A132315
KEYWORD
nonn
STATUS
approved