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

Author

Peter Luschny, Mar 14 2023

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

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

Programs

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 *)

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.