A362715 -id:A362715 - cp's OEIS frontend

A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^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

Stefano Spezia, Apr 30 2023

nonn

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.

Cf. A317615, A362714, A362715.

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]

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.

A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Apr 30 2023

nonn

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.

Cf. A317651, A362713, A362715.

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 *)

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)).

cp's OEIS Frontend

A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

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A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.

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cp's OEIS Frontend

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.

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Mathematica

Formula

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.

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Formula

A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.