A287316 -id:A287316 - cp's OEIS frontend

A336665 a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.

1, 1, 10, 255, 12196, 939155, 106161756, 16554165495, 3404986723720, 893137635015219, 290965846152033460, 115256679181251696803, 54552992572663333862400, 30406695393635479756804525, 19712738332895648545008815416, 14707436666152282009334357074335

Offset: 0

Ilya Gutkovskiy, Jul 29 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..99

Cf. A000275, A033935, A336271, A336638, A336639.

Main diagonal of A340986.

Table[(n!)^2 SeriesCoefficient[1/BesselJ[0, 2 Sqrt[x]]^n, {x, 0, n}], {n, 0, 15}]
A287316[n_, k_] := A287316[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[Binomial[n, j]^2 A287316[n - j, k - 1], {j, 0, n}]]]; b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[(-1)^(j + 1) Binomial[n, j]^2 A287316[j, k] b[n - j, k], {j, 1, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 15}]

a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3.431031961004073074179854315227049823720211... and c = 0.31156343453490677011135864540173577785263... - Vaclav Kotesovec, May 04 2024

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

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jun 24 2025

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

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

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

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);

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

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

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

cp's OEIS Frontend

A336665 a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.

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