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%I #38 Jan 28 2017 00:24:05
%S 1,13,-83,649,-59,2089,-7379,8829,-410479,84273,-4091,2032897,-867947,
%T 951417,-47224023,2228469,-262139,19669687769,-1048571,1461748549,
%U -1500199283,746586657,-16777211,747004180629,-6777994779,7113541809,-13667368865299,29908738140693
%N Numerators of 1/2 + sum(chi(p)^(4*n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4*n and the numbers chi(p) are given by A002172.
%H Seiichi Manyama, <a href="/A274398/b274398.txt">Table of n, a(n) for n = 1..3239</a>
%F The n-th Hurwitz number is A002306(n)/A047817(n) = a(n)/A047817(n) + A002770(n).
%F a(n) = A002306(n) - A002770(n) * A047817(n) for n > 1.
%e H_1 = 1/10 = 1/2 - 2/5 = 1/10, so a(1) = 1.
%e H_2 = 3/10 = 1/2 + 2^2/5 - 1 = 13/10 - 1, so a(2) = 13.
%e H_3 = 567/130 = 1/2 - 2^3/5 + 6/13 + 5 = -83/130 + 5, so a(3) = -83.
%e H_4 = 43659/170 = 1/2 + 2^4/5 + 2/17 + 253 = 649/170 + 253, so a(4) = 649.
%t nmax = 28; H[n_] := (n*(4*n - 2)!/(2^(4*n - 2)))*SeriesCoefficient[ WeierstrassP[z, {4, 0}], {z, 0, 4*n - 2}]; pp = Select[Prime[Range[2 nmax]], Mod[#, 4] == 1 &]; Scan[(chi[#] = -Sum[JacobiSymbol[x^3 - x, #], {x, 0, # - 1}])&, pp]; a[n_] := 1/2 + Sum[If[Divisible[4 n, p - 1], chi[p]^(4*n/(p - 1))/p, 0], {p, pp}] // Numerator; Array[a, nmax] (* _Jean-François Alcover_, Oct 22 2016 *)
%Y Cf. A002172, A002306, A002770, A047817.
%K sign
%O 1,2
%A _Seiichi Manyama_, Sep 26 2016
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