A361522 The aerated factorial numbers.

1, 0, 1, 0, 2, 0, 6, 0, 24, 0, 120, 0, 720, 0, 5040, 0, 40320, 0, 362880, 0, 3628800, 0, 39916800, 0, 479001600, 0, 6227020800, 0, 87178291200, 0, 1307674368000, 0, 20922789888000, 0, 355687428096000, 0, 6402373705728000, 0, 121645100408832000, 0, 2432902008176640000

Offset

0,5

Comments

An aerated version of A000142, which is the main entry for this sequence.

Links

Table of n, a(n) for n=0..40.

Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

Eric Weisstein's World of Mathematics, Error function erf.

Formula

a(n) = n! * [z^n] (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1.

a(n) = n! * [z^n] 1 + 2*u*exp(u)*hypergeom([1/2], [3/2], -u), where u = (z/2)^2.

Maple

egf := (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1:
ser := series(egf, z, 42): seq(n!*coeff(ser, z, n), n = 0..40);

Mathematica

a[n_] := If[OddQ[n], 0, (n/2)!]; Array[a, 41, 0] (* Amiram Eldar, Mar 14 2023 *)

Crossrefs

Cf. A000142, A081124, A084261, A081123, A161556.

Sequence in context: A094233 A094659 A321907 * A137437 A183189 A330609

Adjacent sequences: A361519 A361520 A361521 * A361523 A361524 A361525

Keyword

nonn

Author

Peter Luschny, Mar 14 2023

Status

approved