A394679 Expansion of 2F1(1/4, 3/4; 1/2; 4*x/(1-x)^2)^2.

1, 3, 17, 95, 537, 3059, 17513, 100607, 579377, 3342627, 19311873, 111696991, 646614345, 3745984787, 21714513369, 125937787135, 730719993441, 4241392619843, 24626744856305, 143030575690079, 830920303820921, 4828209352231731, 28060704773688393, 163112737069681663

Offset

0,2

Comments

Integer sequence arising from the symmetric square of a Gauss hypergeometric function composed with a degree-2 rational map phi(x) = 4*x/(1-x)^2. Not an Apery-like sequence: no order-2 polynomial recurrence was found with degree <= 12.

Links

Table of n, a(n) for n=0..23.

Alex Shvets, Integer sequences from Sym^2(2F1) Belyi pullback scan, Zenodo, 2026.

Formula

a(n) = [x^n] 2F1(1/4, 3/4; 1/2; 4*x/(1-x)^2)^2.

Recurrence: (2*n+3)*(n+4)*a(n+4) - 6*(4*n^2+19*n+20)*a(n+3) + 4*(19*n^2+76*n+75)*a(n+2) - 6*(4*n^2+13*n+8)*a(n+1) + n*(2*n+5)*a(n) = 0.

a(n) ~ 2^(-3/2) * (1 + sqrt(2))^(2*n) * (1 + 2^(1/4)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 30 2026

Example

For n=0 the recurrence reads 3*4*a(4) - 6*20*a(3) + 4*75*a(2) - 6*8*a(1) + 0*a(0) = 0, i.e., 12*a(4) = 120*95 - 300*17 + 48*3 = 11400 - 5100 + 144 = 6444, so a(4) = 6444/12 = 537.

Mathematica

CoefficientList[Series[Hypergeometric2F1[1/4, 3/4, 1/2, 4x/(1-x)^2]^2, {x, 0, 23}], x] (* Stefano Spezia, Mar 29 2026 *)

Crossrefs

Cf. A393810, A394680.

Sequence in context: A290925 A020056 A086842 * A151330 A370570 A302871

Adjacent sequences: A394676 A394677 A394678 * A394680 A394681 A394682

Keyword

nonn

Author

Alex Shvets, Mar 28 2026

Extensions

a(20)-a(23) from Stefano Spezia, Mar 29 2026

Status

approved