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A275586
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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)!).
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3
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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
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OFFSET
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1,2
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COMMENTS
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Integers that do not appear uniquely in the Catalan triangle A009766.
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LINKS
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EXAMPLE
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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
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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