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A091043
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Normalized triangle of odd numbered entries of even numbered rows of Pascal's triangle A007318.
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3
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1, 1, 1, 3, 10, 3, 1, 7, 7, 1, 5, 60, 126, 60, 5, 3, 55, 198, 198, 55, 3, 7, 182, 1001, 1716, 1001, 182, 7, 1, 35, 273, 715, 715, 273, 35, 1, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 5, 285, 3876, 19380, 41990, 41990, 19380, 3876, 285, 5, 11, 770, 13167, 85272
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| b(n)= A006519(n), with b(n) defined in the formula. For every odd n b(n)=1.
The row polynomials Po(n,x) := 2*b(n)*sum(a(n,m)*x^m,m=0..n-1), n>=1, appear as numerators of the generating functions for the odd numbered column sequences of array A034870. b(n) is defined in the formula below.
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LINKS
| W. Lang, First 9 rows.
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FORMULA
| a(n, m)= binomial(2*n, 2*m+1)/(2*b(n)), n>=m+1>=1, else 0, with b(n) := GCD(seq(binomial(2*n, 2*m+1)/2, m=0..n-1)), where GCD denotes the greatest common divisor of a set of numbers (here one half of the odd numbered entries in the even numbered rows of Pascal's triangle). It suffices to consider m=0..floor((n-1)/2) due to symmetry.
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EXAMPLE
| [1];[1,1];[3,10,3];[1,7,7,1];[5,60,126,60,5];...
n=3: GCD(3,10,3)=GCD(3,10)=1=b(3)=A006519(3); n=4: GCD(4,28,28,4)=GCD(4,28)=4=b(4)=A006519(4).
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CROSSREFS
| Sequence in context: A179397 A111272 A124692 * A167790 A010708 A072988
Adjacent sequences: A091040 A091041 A091042 * A091044 A091045 A091046
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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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