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%I #27 Nov 20 2019 21:40:27
%S 1,-72,48384,-134120448,1055796166656,-18987644270149632,
%T 676784742282773397504,-43249455805185586718834688,
%U 4599203617006025540525554139136,-768291761151281123722697889747566592,192565676807771292904270021964021234663424
%N Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.
%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] See Eq. (16) and Table III.
%H Tanay Wakhare, 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 Hurwitz (Eq. (84)) gives a recurrence.
%F a(n) = (-12)^n * A144849(n). - _R. J. Mathar_, Aug 03 2015
%p A260779 := proc(n)
%p option remember;
%p if n = 0 then
%p 1;
%p else
%p a :=0 ;
%p for r from 0 to n-1 do
%p s := n-1-r ;
%p if s >=0 and s <= n-1 then
%p a := a+procname(r)*procname(s) *binomial(4*n,4*r+2) ;
%p end if;
%p end do:
%p a*(-12) ;
%p end if;
%p end proc: # _R. J. Mathar_, Aug 03 2015
%t 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 *)
%Y Cf. A144849.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_, Aug 02 2015
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