A395124 Expansion of 2F1(1/3, 2/3; 1; 27*x/(1-4*x)^3)^2.

1, 12, 360, 10776, 337656, 10931616, 362216088, 12210185424, 417084184152, 14397169640160, 501213389053440, 17572037287514688, 619710909219471000, 21965689397233570944, 781966883434618883664, 27943249164113261680992, 1001870741488022086850328, 36026988854859239051323296

Offset

0,2

Comments

Integer sequence arising from the symmetric square of a Gauss hypergeometric function composed with a degree-3 Belyi map phi(x) = 27*x/(1-4*x)^3. Not an Apery-like sequence: no order-2 polynomial recurrence was found with degree <= 12.

Links

Table of n, a(n) for n=0..17.

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

Formula

a(n) = [x^n] 2F1(1/3, 2/3; 1; 27*x/(1-4*x)^3)^2.

Recurrence: n^3*(3*n-10)*(3*n-8)*(3*n-7)*(243*n^6 - 4617*n^5 + 35721*n^4 - 143163*n^3 + 310500*n^2 - 341676*n + 148720)*a(n)=3*(3*n-10)*(56862*n^11 - 1449981*n^10 + 16222923*n^9 - 104648598*n^8 + 430153092*n^7 - 1176229125*n^6 + 2171201895*n^5 - 2693917800*n^4 + 2199585260*n^3 - 1132156800*n^2 + 336053952*n-44413440)*a(n-1) - 3*(n-2)*(3536379*n^11 - 100197405*n^10 + 1261129176*n^9 - 9283602618*n^8 + 44268484095*n^7 - 143032520061*n^6 + 318142103382*n^5 - 484798038444*n^4 + 493484544216*n^3 - 318035920512*n^2 + 116426392704*n-18393533440)*a(n-2) + 16*(1587762*n^12 - 50543757*n^11 + 723390345*n^10 - 6145093836*n^9 + 34431657228*n^8 - 133669124289*n^7 + 367331146029*n^6 - 716858252622*n^5 + 981167167884*n^4 - 913791988120*n^3 + 547107273216*n^2 - 188622251136*n + 28445967360)*a(n-3) - 384*(124659*n^12 - 4155300*n^11 + 61993431*n^10 - 546240186*n^9 + 3158070849*n^8 - 12582221940*n^7 + 35286189849*n^6 - 69849292518*n^5 + 96302439516*n^4 - 89594845960*n^3 + 53042224320*n^2 - 17872121472*n + 2609157120)*a(n-4) + 3072*(n-5)*(3*n-4)*(4374*n^10 - 124659*n^9 + 1546938*n^8 - 10980360*n^7 + 49188384*n^6 - 144470925*n^5 + 279348624*n^4 - 346986504*n^3 + 261089952*n^2 - 106116624*n + 17971840)*a(n-5) - 4096*(n-6)^3*(3*n-7)*(3*n-5)*(3*n-4)*(243*n^6 - 3159*n^5 + 16281*n^4 - 41589*n^3 + 52812*n^2 - 28908*n + 5728)*a(n-6). - Vaclav Kotesovec, Apr 19 2026

a(n)^(1/n) ~ (1 + (1 + sqrt(2))^(2/3) + 1/(1 + sqrt(2))^(2/3))^3. - Vaclav Kotesovec, Apr 19 2026

Mathematica

CoefficientList[Series[Hypergeometric2F1[1/3, 2/3, 1, 27*x/(1-4*x)^3]^2, {x, 0, 17}], x]

Crossrefs

Cf. A393810, A394679, A394680.

Sequence in context: A012386 A130213 A012633 * A009079 A012379 A012424

Adjacent sequences: A395120 A395122 A395123 * A395125 A395126 A395127

Keyword

nonn

Author

Alex Shvets, Apr 12 2026

Status

approved