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A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

%I A260779 #33 Jul 12 2024 10:17:35
%S A260779 1,-72,48384,-134120448,1055796166656,-18987644270149632,
%T A260779 676784742282773397504,-43249455805185586718834688,
%U A260779 4599203617006025540525554139136,-768291761151281123722697889747566592,192565676807771292904270021964021234663424
%N A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.
%C A260779 This is for the lemniscate case where g2=4, g3=0. - _Michael Somos_, Jul 10 2024
%H A260779 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] See Eq. (16) and Table III.
%H A260779 Tanay Wakhare and Christophe Vignat, <a href="https://arxiv.org/abs/1909.01508">Taylor coefficients of the Jacobi theta3(q) function</a>, arXiv:1909.01508 [math.NT], 2019.
%F A260779 Hurwitz (Eq. (84)) gives a recurrence.
%F A260779 a(n) = (-12)^n * A144849(n). - _R. J. Mathar_, Aug 03 2015
%p A260779 A260779 := proc(n)
%p A260779     option remember;
%p A260779     if n = 0 then
%p A260779         1;
%p A260779     else
%p A260779         a :=0 ;
%p A260779         for r from 0 to n-1 do
%p A260779             s := n-1-r ;
%p A260779             if s >=0 and s <= n-1 then
%p A260779             a := a+procname(r)*procname(s) *binomial(4*n,4*r+2) ;
%p A260779             end if;
%p A260779         end do:
%p A260779         a*(-12) ;
%p A260779     end if;
%p A260779 end proc: # _R. J. Mathar_, Aug 03 2015
%t A260779 Block[{a}, a[n_] := If[n < 1, Boole[n == 0], Sum[Binomial[4 n, 4 j + 2] a[j] a[n - 1 - j], {j, 0, n - 1}]]; Array[(-12)^#*a[#] &, 11, 0]] (* _Michael De Vlieger_, Nov 20 2019, after _Harvey P. Dale_ at A144849 *)
%t A260779 a[ n_] := If[n<0, 0, With[{m = 4*n+2}, m!/2*SeriesCoefficient[ 1/WeierstrassP[u, {4, 0}], {u, 0, m}]]]; (* _Michael Somos_, Jul 10 2024 *)
%Y A260779 Cf. A144849.
%K A260779 sign,easy
%O A260779 0,2
%A A260779 _N. J. A. Sloane_, Aug 02 2015