cp's OEIS Frontend

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).

%I A362715 #10 May 01 2023 10:02:25
%S A362715 1,0,1,0,48,1,0,7584,240,1,0,2515968,97664,672,1,0,1432498176,
%T A362715 63221760,560448,1440,1,0,1247557386240,60299053056,628024320,2141568,
%U A362715 2640,1,0,1542446268088320,79885647249408,933093697536,3819239424,6374368,4368,1
%N 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).
%H A362715 Christian Krattenthaler and Thomas W. Müller, <a href="https://arxiv.org/abs/2304.11471">The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3</a>, arXiv:2304.11471 [math.NT], 2023. See p. 6.
%e A362715 The triangle begins:
%e A362715     1;
%e A362715     0,          1;
%e A362715     0,         48,        1;
%e A362715     0,       7584,      240,      1;
%e A362715     0,    2515968,    97664,    672,    1;
%e A362715     0, 1432498176, 63221760, 560448, 1440, 1;
%e A362715     ...
%t A362715 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
%Y A362715 Cf. A317651, A362713, A362714.
%K A362715 nonn,tabl
%O A362715 0,5
%A A362715 _Stefano Spezia_, Apr 30 2023