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A124399
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Numerators of one half of norm square of monic Legendre polynomials p_n(x).
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1
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1, 1, 4, 4, 64, 64, 256, 256, 16384, 16384, 65536, 65536, 1048576, 1048576, 4194304, 4194304, 1073741824, 1073741824, 4294967296, 4294967296, 68719476736, 68719476736, 274877906944, 274877906944, 17592186044416, 17592186044416
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Denominators are given by A069955.
The rationals N2(n):=2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [ -1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference
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REFERENCES
| R. Courant, D. Hilbert, Methoden der mathematischen Physik, Bd.I, 3. Auflage, Springer, pp. 73-4.
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LINKS
| W. Lang: Norm square, rationals and more.
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FORMULA
| a(n)=numerator(N2(n)/2) with N2(n)/2:=(1/(2*n+1))*((2^n)/binomial(2*n,n))^2.
N2(n)/2= (1/(2*n+1))*(1/L(n))^2 with L(n)= A001790(n)/A060818(n), the leading coefficient of the Legendre polynomial P_n(x), in lowest terms.
Bisection: a(2*n)=a(2*n+1)= A056982(n), n>=0.
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EXAMPLE
| Rationals a(n)/A069955(n): [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, ...].
Rationals N2(n): [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693,...].
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CROSSREFS
| Sequence in context: A009644 A196180 A156483 * A119600 A107053 A068376
Adjacent sequences: A124396 A124397 A124398 * A124400 A124401 A124402
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 10 2006
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