A274398 - OEIS

A274398

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.

1, 13, -83, 649, -59, 2089, -7379, 8829, -410479, 84273, -4091, 2032897, -867947, 951417, -47224023, 2228469, -262139, 19669687769, -1048571, 1461748549, -1500199283, 746586657, -16777211, 747004180629, -6777994779, 7113541809, -13667368865299, 29908738140693

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OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..3239

FORMULA

The n-th Hurwitz number is A002306(n)/A047817(n) = a(n)/A047817(n) + A002770(n).

a(n) = A002306(n) - A002770(n) * A047817(n) for n > 1.

EXAMPLE

H_1 = 1/10 = 1/2 - 2/5 = 1/10, so a(1) = 1.

H_2 = 3/10 = 1/2 + 2^2/5 - 1 = 13/10 - 1, so a(2) = 13.

H_3 = 567/130 = 1/2 - 2^3/5 + 6/13 + 5 = -83/130 + 5, so a(3) = -83.

H_4 = 43659/170 = 1/2 + 2^4/5 + 2/17 + 253 = 649/170 + 253, so a(4) = 649.

MATHEMATICA

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 *)

CROSSREFS

Cf. A002172, A002306, A002770, A047817.

Sequence in context: A142085 A376916 A163688 * A301953 A321589 A225129

Adjacent sequences: A274395 A274396 A274397 * A274399 A274400 A274401

KEYWORD

sign

AUTHOR

Seiichi Manyama, Sep 26 2016

STATUS

approved