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Search: id:A002306
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| A002306 |
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Numerators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).
(Formerly M3179 N1288)
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+0
4
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| 1, 3, 567, 43659, 392931, 1724574159, 2498907956391, 1671769422825579, 88417613265912513891, 21857510418232875496803, 2296580829004860630685299, 3133969138162958884235052785487, 6456973729353591041508572318079423
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
L. Carlitz, The coefficients of the lemniscate function, Math. Comp., 16 (1962), 475-478.
F. Lemmermeyer, Reciprocity Laws, Springer-Veralg, 2000; see p. 276.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..60
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FORMULA
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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
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Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ... = A002306/A047817.
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MAPLE
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A002306 := 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)*A002306(k)*A002306(n - k), k = 1 .. n - 1)/( (2*n - 3)*(16*n^2 - 1)) fi; end;
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PROGRAM
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(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
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Denominators give A047817.
Sequence in context: A100339 A158247 A034314 this_sequence A087574 A153402 A121043
Adjacent sequences: A002303 A002304 A002305 this_sequence A002307 A002308 A002309
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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