A002770 Integers connected with coefficients in expansion of Weierstrass P-function.

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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]

Crossrefs

Cf. A002172, A002306, A047817.

Programs

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 *)

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

Extensions

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