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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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4
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1, 3, 567, 43659, 392931, 1724574159, 2498907956391, 1671769422825579, 88417613265912513891, 21857510418232875496803, 2296580829004860630685299, 3133969138162958884235052785487, 6456973729353591041508572318079423
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OFFSET
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1,2
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REFERENCES
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L. Carlitz, The coefficients of the lemniscate function, Math. Comp., 16 (1962), 475-478.
F. Lemmermeyer, Reciprocity Laws, Springer-Veralg, 2000; see p. 276.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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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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; A002306 := n -> numer(H(n)); seq(A002306(n), n=1..15);
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MATHEMATICA
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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}]; Numerator[ Table[a[n], {n, 1, 13}]] (* From Jean-François Alcover, Oct 18 2011, after PARI *)
p[z_] := WeierstrassP[z, {4, 0}]; se = Series[ p[z] - 1/z^2 - Sum[ 2^(4*n)*h[n]*z^(4*n-2) / (4*n*(4*n-2)!), {n, 1, 13}], {z, 0, 4*13}] // Normal; Array[h, 13] /. SolveAlways[se == 0, z][[1]] // Numerator (* Jean-François Alcover, Sep 07 2012 *)
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PROG
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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: A034314 A201431 A171359 * A087574 A153402 A121043
Adjacent sequences: A002303 A002304 A002305 * 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.
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STATUS
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approved
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