OFFSET
1,2
COMMENTS
Named after the German mathematician Adolf Hurwitz (1859-1919). - Amiram Eldar, Jun 24 2021
REFERENCES
Franz Lemmermeyer, Reciprocity Laws, Springer-Verlag, 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).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..152 (first 60 terms from T. D. Noe)
L. Carlitz, The coefficients of the lemniscate function, Math. Comp., Vol. 16, No. 80 (1962), pp. 475-478.
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., Vol. 51 (1899), pp. 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy]
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} 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.
EXAMPLE
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; A002306 := n -> numer(H(n)); seq(A002306(n), n=1..15);
# Alternative program
c := n -> (n*(4*n-2)!/(2^(4*n-2)))*coeff(series(WeierstrassP(z, 4, 0), z, 4*n+2), z, 4*n-2); a := n -> numer(c(n)); seq(a(n), n=1..13); # Peter Luschny, Aug 18 2014
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}]; Numerator[ Table[a[n], {n, 1, 13}]] (* Jean-François Alcover, Oct 18 2011, after PARI *)
p[z_] := WeierstrassP[z, {4, 0}]; a[n_] := (n*(4*n-2)!/(2^(4*n-2))) * SeriesCoefficient[p[z], {z, 0, 4*n-2}] // Numerator; Array[a, 13] (* Jean-François Alcover, Sep 07 2012, updated Oct 22 2016 *)
a[ n_] := If[ n < 0, 0, Numerator[ 2^(-4 n) (4 n)! SeriesCoefficient[ 1 - x WeierstrassZeta[ x, {4, 0}], {x, 0, 4 n}]]]; (* Michael Somos, Mar 05 2015 *)
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])) \\ Henri Cohen, Mar 18 2002
CROSSREFS
KEYWORD
nonn,easy,nice,frac
AUTHOR
STATUS
approved