A362713 Expansion of e.g.f. x*2F1([3/4, 3/4], [3/2], 4*x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

1, 6, 256, 28560, 6071040, 2098483200, 1071889920000, 758870167910400, 711206089850880000, 852336059876720640000, 1271438437097485762560000, 2310211006286602237378560000, 5023141810386294125321256960000, 12877606625796048169971744768000000, 38439740210093310755176533983232000000

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. 5.

Crossrefs

Cf. A317615, A362714, A362715.

Programs

Mathematica

Table[(2n+1)!SeriesCoefficient[x*Hypergeometric2F1[3/4, 3/4, 3/2, 4*x^2]/Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2], {x, 0, 2n+1}], {n,0, 14}]
(* or *)
a[0]=1; a[n_]:=Product[(4j-1)^2,{j,n}]-Sum[Binomial[2n+1,2m+1]Product[(4j-3)^2,{j,n-m}]a[m],{m,0,n-1}]; Array[a,15,0]

Formula

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