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).

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

0,5

Links

Table of n, a(n) for n=0..35.

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.

Example

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

Mathematica

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

Crossrefs

Cf. A317651, A362713, A362714.

Sequence in context: A002834 A057380 A234565 * A036210 A257875 A037941

Adjacent sequences: A362712 A362713 A362714 * A362716 A362717 A362718

Keyword

nonn,tabl

Author

Stefano Spezia, Apr 30 2023

Status

approved