%I #13 May 03 2023 09:24:19

%S 1,1,47,7395,2453425,1399055625,1221037941375,1513229875486875,

%T 2526879997358510625,5469272714829657020625,

%U 14892997153152592003359375,49826568404835717359311321875,200913471834337931507493300140625,960945974809003219596852282787265625,5378917217051713436481068409370884609375

%N a(0) = 1 and a(n) = 2^(n-1)*Product_{j=1..n} (4*j - 3)^2 - Sum_{m=1..n-1} binomial(2*n, 2*m)*a(m)*a(n-m)/2 for n > 0.

%H Christian Krattenthaler and Thomas W. Müller, <a href="https://arxiv.org/abs/2304.11471">The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3</a>, arXiv:2304.11471 [math.NT], 2023. See p. 6.

%F E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2^n*(2*n)!) = sqrt(2F1([1/4, 1/4], [1/2], 4*x^2)).

%t a[0]=1; a[n_]:=2^(n-1)Product[(4j-3)^2,{j,n}]-Sum[Binomial[2n,2m]a[m]a[n-m],{m,n-1}]/2; Array[a,15,0]

%t nmax = 20; Table[(k-1)! * 2^((k-1)/2) * CoefficientList[Series[Sqrt[Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2]], {x, 0, 2*nmax+2}], x][[k]], {k, 1, 2*nmax+2, 2}] (* _Vaclav Kotesovec_, May 03 2023 *)

%Y Cf. A317651, A362713, A362715.

%K nonn

%O 0,3

%A _Stefano Spezia_, Apr 30 2023