A362714 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.

1, 1, 47, 7395, 2453425, 1399055625, 1221037941375, 1513229875486875, 2526879997358510625, 5469272714829657020625, 14892997153152592003359375, 49826568404835717359311321875, 200913471834337931507493300140625, 960945974809003219596852282787265625, 5378917217051713436481068409370884609375

Offset: 0

Author

Stefano Spezia, Apr 30 2023

Keywords

• nonn

Links

• Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 6.

Crossrefs

Cf. A317651, A362713, A362715.

Programs

Mathematica

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]
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 *)

Formula

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)).