A034177 a(n) is the n-th quartic factorial number divided by 4.

1, 8, 96, 1536, 30720, 737280, 20643840, 660602880, 23781703680, 951268147200, 41855798476800, 2009078326886400, 104472072998092800, 5850436087893196800, 351026165273591808000, 22465674577509875712000, 1527665871270671548416000, 109991942731488351485952000

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..365

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 513.

Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Formula

4*a(n) = (4*n)(!^4) = Product_{j=1..n} 4*j = 4^n * n!.

E.g.f.: (-1 + 1/(1-4*x))/4.

D-finite with recurrence: a(n) -4*n*a(n-1)=0. - R. J. Mathar, Feb 24 2020

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 4*(exp(1/4)-1).

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(1-exp(-1/4)). (End)

Example

G.f. = x + 8*x^2 + 96*x^3 + 1536*x^4 + 30720*x^5 + 737820*x^6 + ...

Maple

[seq(n!*4^(n-1), n=1..16)]; # Zerinvary Lajos, Sep 23 2006

Mathematica

Array[4^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

Prog

(PARI) vector(20, n, 4^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019
(Magma) [4^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019
(Sage) [4^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019
(GAP) List([1..20], n-> 4^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 15 2019

Crossrefs

Cf. A007696, A000407, A034176. First column of triangle A048786.

A052570 is an essentially identical sequence. - Philippe Deléham, Sep 18 2008

Equals the second right hand column of A167569 divided by 2. - Johannes W. Meijer, Nov 12 2009

Sequence in context: A369538 A220285 A052570 * A002168 A114425 A224767

Adjacent sequences: A034174 A034175 A034176 * A034178 A034179 A034180

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved