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A091044
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One half of odd numbered entries of even numbered rows of Pascal's triangle A007318.
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3
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1, 2, 2, 3, 10, 3, 4, 28, 28, 4, 5, 60, 126, 60, 5, 6, 110, 396, 396, 110, 6, 7, 182, 1001, 1716, 1001, 182, 7, 8, 280, 2184, 5720, 5720, 2184, 280, 8, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 11
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The odd numbered columns of this triangle can be reduced: see triangle A091043.
The odd numbered rows coincide with the ones of the reduced triangle A091043.
binomial(2*n,2*m+1) is even for n>=m+1>=1, hence every T(n,m) is a positive integer.
The GCD (greatest common divisor) of the entries of each odd numbered row n=2*k+1, k>=0, is 1.
The GCD of the entries of the even numbered row n=2*k, k>=1, is A006519(n) (highest power of 2 in n=2*k).
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LINKS
| W. Lang, First 9 rows.
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FORMULA
| T(n, m)= binomial(2*n, 2*m+1)/2, n>=m+1>=1, else 0.
Put a(n) = n!*(n+1/2)!/(1/2)!. T(n+1,k) = (n+1)*a(n)/(a(k)*a(n-k)).
T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1). Cf. A111910. - Peter Bala Oct 13 2011
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EXAMPLE
| [1];[2,2];[3,10,3];[4,28,28,4];[5,60,126,60,5];[6,110,396,396,110,6];...
n=6=2*3: GCD(6,110,396)=2=A006519(6); n=5:
GCD(5,60,126)=1=A006519(5).
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CROSSREFS
| A111910.
Sequence in context: A019234 A032172 A032103 * A079661 A153920 A067579
Adjacent sequences: A091041 A091042 A091043 * A091045 A091046 A091047
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 23 2004
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