A002770 Integers connected with coefficients in expansion of Weierstrass P-function. (Formerly M4033 N1675)

-1, 5, 253, 39299, 13265939, 8616924013, 9833937781275, 18382040180023477, 53311001020080183933, 229658082900486063068939, 1418085582879166915943461879, 12182969300667152908506740224429, 141998788870155117956738989275999795

Offset

2,2

References

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 = 2..152

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]

Formula

a(n) = A002306(n) / A047817(n) - 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. The resulting a(n) is an integer despite all the rationals. - Sean A. Irvine, Aug 17 2014

Mathematica

nmax = 20; 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_] := H[n] - 1/2 - Sum[If[Divisible[4 n, p - 1], chi[p]^(4*n/(p - 1))/p, 0], {p, pp}]; Table[a[n], {n, 2, nmax}] (* Jean-François Alcover, Dec 11 2014, updated Oct 22 2016 *)

Crossrefs

Cf. A002172, A002306, A047817.

Sequence in context: A336295 A332125 A308652 * A069071 A275930 A181397

Adjacent sequences: A002767 A002768 A002769 * A002771 A002772 A002773

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 07 2004

Status

approved