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%I #63 Apr 04 2023 11:05:06
%S 10,10,130,170,10,130,290,170,4810,410,10,2210,530,290,7930,170,10,
%T 351130,10,6970,3770,890,10,214370,1010,530,524290,557090,10,325130,
%U 10,170,130,1370,290,5969210,1490,10,1081730,6970,10,3770
%N Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).
%C 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
%D F. Lemmermeyer, Reciprocity Laws, Springer-Verlag, 2000; see p. 276.
%H T. D. Noe, <a href="/A047817/b047817.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0152678-2">The coefficients of the lemniscate function</a>, Math. Comp., 16 (1962), 475-478.
%H A. Hurwitz, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1897&DMDID=DMDLOG_0041&LOGID=LOG_0041&PHYSID=PHYS_0289">Über die Entwicklungskoeffizienten der lemniskatischen Funktionen</a>, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
%H A. Hurwitz, <a href="/A002306/a002306.pdf">Über die Entwicklungskoeffizienten der lemniskatischen Funktionen</a>, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy]
%H N. M. Katz, <a href="https://web.math.princeton.edu/~nmk/old/congKlvS.pdf">The Congruences of Clausen - von Staudt and Kummer for Bernoulli-Hurwitz Numbers</a>, Mathematische Annalen 216, 1-4 (1975)
%H Alexei Pantchichkine, <a href="https://hal.archives-ouvertes.fr/hal-01875160">Constructions of p-adic L-functions and admissible measures for Hermitian modular forms</a>, Number Theory [math.NT], 2018.
%H Alexei Pantchichkine, <a href="https://science.thanglong.edu.vn/index.php/volc/article/view/39">Algebraic differential operators on arithmetic automorphic forms, modular distributions, p-adic interpolation of their critical l values via BGG modules and Hecke algebras</a>, J. Math. Math. Sci., Thang Long Univ. (Viet Nam, 2022) Vol. 1, No. 4, 1-26.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F 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)!).
%F 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)).
%F See PARI line for recurrence.
%e Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ...
%p 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;
%p a := n -> denom(H(n));
%p # Implementation based on Hurwitz's extension of Clausen's theorem:
%p GenClausen := proc(n) local k,S; map(k->k+1, numtheory[divisors](n));
%p S := select(p-> isprime(p) and p mod 4 = 1, %);
%p if S <> {} then 2*mul(k,k=S) else NULL fi end:
%p A047817_list := proc(n) local i; seq(GenClausen(i),i=1..4*n) end;
%p A047817_list(42); # _Peter Luschny_, Oct 03 2011
%p # Implementation based on Weierstrass's P-function:
%p 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
%t 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 *)
%t 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 *)
%o (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
%Y For numerators see A002306.
%Y Cf. A160014.
%K nonn,easy,nice,frac
%O 1,1
%A _N. J. A. Sloane_
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