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A047817
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Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).
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3
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10, 10, 130, 170, 10, 130, 290, 170, 4810, 410, 10, 2210, 530, 290, 7930, 170, 10, 351130, 10, 6970, 3770, 890, 10, 214370, 1010, 530, 524290, 557090, 10, 325130, 10, 170, 130, 1370, 290, 5969210, 1490, 10, 1081730, 6970, 10, 3770
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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REFERENCES
| F. Lemmermeyer, Reciprocity Laws, Springer-Veralg, 2000; see p. 276.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
L. Carlitz, The coefficients of the lemniscate function, Math. Comp., 16 (1962), 475-478.
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FORMULA
| Let P be the Weierstrass P-function satisfying P'^2 = 4*P^3 - 4*P. Then P(z) = 1/z^2 + Sum_{n=1..infinity} 2^(4n)*H_n*z^(4n-2)/(4n*(4n-2)!).
Sum_{ (r, s) != (0, 0) } 1/(r+si)^(4n) = (2w)^(4n)*H_n/(4n)! where w = 2 * Integral_{0..1} dx/(sqrt(1-x^4)).
See PARI line for recurrence.
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EXAMPLE
| Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ...
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MAPLE
| H := proc(n) local k; option remember; if n = 1 then 1/10 else 3*add((4*k - 1)*(4*n - 4*k - 1)*binomial(4*n, 4*k)*H(k)*H(n - k), k = 1 .. n - 1)/( (2*n - 3)*(16*n^2 - 1)) fi; end;
a := n -> denom(H(n));
# implementation by Peter Luschny, Oct 03 2011:
GenClausen := proc(n) local k, S; map(k->k+1, numtheory[divisors](n));
S := select(p-> isprime(p) and p mod 4 = 1, %);
if S <> {} then 2*mul(k, k=S) else NULL fi
end:
A047817_list := proc(n) local i; seq(GenClausen(i), i=1..4*n) end;
A047817_list(42); # Peter Luschny, Oct 03 2011
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MATHEMATICA
| a[1] = 1/10; a[n_] := a[n] = (3/(2*n - 3)/(16*n^2 - 1))* Sum[(4*k - 1)*(4*n - 4*k - 1)*Binomial[4*n, 4*k]*a[k]* a[n - k], {k, 1, n - 1}]; Denominator[ Table[a[n], {n, 1, 42}]] (* From Jean-François Alcover, Oct 18 2011, after PARI *)
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PROG
| (PARI) do(lim)=v=vector(lim); v[1]=1/10; for(n=2, lim, v[n]=3/(2*n-3)/(16*n^2-1)*sum(k=1, n-1, (4*k-1)*(4*n-4*k-1)*binomial(4*n, 4*k)*v[k]*v[n-k])) - from Henri Cohen, Mar 18, 2002
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CROSSREFS
| For numerators see A002306.
Cf. A160014.
Sequence in context: A165428 A004286 A145595 * A065243 A105750 A052983
Adjacent sequences: A047814 A047815 A047816 * A047818 A047819 A047820
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KEYWORD
| nonn,easy,nice,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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