A361522 - OEIS

%I #16 Nov 14 2023 13:42:43

%S 1,0,1,0,2,0,6,0,24,0,120,0,720,0,5040,0,40320,0,362880,0,3628800,0,

%T 39916800,0,479001600,0,6227020800,0,87178291200,0,1307674368000,0,

%U 20922789888000,0,355687428096000,0,6402373705728000,0,121645100408832000,0,2432902008176640000

%N The aerated factorial numbers.

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

%H Sebastian Volz, <a href="https://www.uni-saarland.de/fileadmin/upload/lehrstuhl/weber-moritz/Abschlussarbeiten/volz-bachelors-thesis.pdf">Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets</a>, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erf.html">Error function erf</a>.

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

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

%p egf := (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1:

%p ser := series(egf, z, 42): seq(n!*coeff(ser, z, n), n = 0..40);

%t a[n_] := If[OddQ[n], 0, (n/2)!]; Array[a, 41, 0] (* _Amiram Eldar_, Mar 14 2023 *)

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

%K nonn

%O 0,5

%A _Peter Luschny_, Mar 14 2023