OFFSET
1,1
COMMENTS
Hurwitz showed (see Katz, eqn. 9) that a(n) = product of the prime p = 2 and the primes p of the form 4*k + 1 such that p - 1 divides 4*n. It follows that a(n) is a divisibility sequence, that is, if n | m then a(n) | a(m). - Peter Bala, Jan 08 2014
REFERENCES
F. Lemmermeyer, Reciprocity Laws, Springer-Verlag, 2000; see p. 276.
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.
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.
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]
N. M. Katz, The Congruences of Clausen - von Staudt and Kummer for Bernoulli-Hurwitz Numbers, Mathematische Annalen 216, 1-4 (1975)
Alexei Pantchichkine, Constructions of p-adic L-functions and admissible measures for Hermitian modular forms, Number Theory [math.NT], 2018.
Alexei Pantchichkine, Algebraic differential operators on arithmetic automorphic forms, modular distributions, p-adic interpolation of their critical l values via BGG modules and Hecke algebras, J. Math. Math. Sci., Thang Long Univ. (Viet Nam, 2022) Vol. 1, No. 4, 1-26.
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
Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ...
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 based on Hurwitz's extension of Clausen's theorem:
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
# Implementation based on Weierstrass's P-function:
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 -> denom(c(n)); seq(a(n), n=1..42); # 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}]; Denominator[ Table[a[n], {n, 1, 42}]] (* Jean-François Alcover, Oct 18 2011, after PARI *)
a[ n_] := If[ n < 1, 0, Denominator[ 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