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%I M4033 N1675 #44 Feb 26 2018 00:53:26
%S -1,5,253,39299,13265939,8616924013,9833937781275,18382040180023477,
%T 53311001020080183933,229658082900486063068939,
%U 1418085582879166915943461879,12182969300667152908506740224429,141998788870155117956738989275999795
%N Integers connected with coefficients in expansion of Weierstrass P-function.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A002770/b002770.txt">Table of n, a(n) for n = 2..152</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]
%F 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
%t 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 *)
%Y Cf. A002172, A002306, A047817.
%K sign,easy,nice
%O 2,2
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 07 2004
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