A385286 a(n) = (n!)^2 [x^n] hypergeom([], [1], x)^8.

1, 8, 120, 2528, 66424, 2039808, 70283424, 2643158400, 106391894904, 4518833256512, 200396211454720, 9205443151733760, 435368682010660000, 21100379936684418560, 1044115187294444772480, 52597451834668445910528, 2691037806733052553149304, 139567074682665782246950080

Offset: 0

Author

Peter Luschny, Jun 24 2025

Keywords

• nonn
• walk

Comments

We regard this sequence in the list of sequences n -> A287316(n, 2^k) for k = 3.

Links

• Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.

Crossrefs

Cf. A000012 (k=0), A000984 (k=1), A002895 (k=2), this sequence (k=3), A287316.

Programs

Maple

A385286_list := proc(len) local n; series(hypergeom([], [1], x)^8, x, len);
seq((n!)^2*coeff(%, x, n), n = 0..len-1) end: A385286_list(18);

Mathematica

nmax = 20; CoefficientList[Series[BesselI[0, 2*Sqrt[x]]^8, {x, 0, nmax}], x] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Jun 24 2025 *)

PARI

a(n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^8, n); \\ Michel Marcus, Jun 24 2025

Formula

a(n) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^8.

a(n) = A287316(n, 2^3).

a(n) ~ 2^(6*n+5) / (Pi^(7/2) * n^(7/2)). - Vaclav Kotesovec, Jun 24 2025