A362714 -id:A362714 - 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.

A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

Original entry on oeis.org

1, 0, 1, 0, 48, 1, 0, 7584, 240, 1, 0, 2515968, 97664, 672, 1, 0, 1432498176, 63221760, 560448, 1440, 1, 0, 1247557386240, 60299053056, 628024320, 2141568, 2640, 1, 0, 1542446268088320, 79885647249408, 933093697536, 3819239424, 6374368, 4368, 1

Offset: 0

Stefano Spezia, Apr 30 2023

			The triangle begins:
    1;
    0,          1;
    0,         48,        1;
    0,       7584,      240,      1;
    0,    2515968,    97664,    672,    1;
    0, 1432498176, 63221760, 560448, 1440, 1;
    ...

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, A362714.

U[x_]:=x*Hypergeometric2F1[3/4, 3/4, 3/2, 4*x^2]/Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2]; T[n_,k_]:=2^(n-k)(2n)!/(2k)!SeriesCoefficient[U[x]^(2k),{x,0,2n}]; Table[T[n,k],{n,0,7},{k,0,n}]//Flatten

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A362715 Triangle read by rows: T(n, k) = 2^(n-k)*(2*n)!/(2*k)! * [x^(2*n)] U[x]^(2*k), where U(x) = x*2F1([3/4, 3/4], [3/2], 4*x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).