A052676 Expansion of e.g.f. 3*x/(1 - 2*x).

0, 3, 12, 72, 576, 5760, 69120, 967680, 15482880, 278691840, 5573836800, 122624409600, 2942985830400, 76517631590400, 2142493684531200, 64274810535936000, 2056793937149952000, 69930993863098368000, 2517515779071541248000, 95665599604718567424000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..350

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 624.

Formula

E.g.f.: 3*x/(1 - 2*x).

Recurrence: a(0) = 0, a(1) = 3, a(n) = 2*n*a(n-1).

a(n) = 3*2^(n-1)*n! for n > 0.

G.f.: (3/2)*(Hypergeometric2F0([1, 1], [], 2*x) - 1). - G. C. Greubel, Jun 12 2022

a(n) = HypergeometricRegularized([1, -n], [2 - n], -2) for n > 0. - Peter Luschny, Apr 26 2024

Maple

spec := [S, {S=Prod(Sequence(Union(Z, Z)), Union(Z, Z, Z))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

With[{nn=20}, CoefficientList[Series[(3x)/(1-2x), {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Feb 23 2013 *)
a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -2];
Join[{0}, Table[a[n], {n, 1, 20}]]  (* Peter Luschny, Apr 26 2024 *)

Prog

(Magma) [0] cat [3*Factorial(n)*2^(n-1): n in [1..30]]; // G. C. Greubel, Jun 12 2022
(SageMath) [(3/2)*(factorial(n)*2^n -bool(n==0)) for n in (0..30)] # G. C. Greubel, Jun 12 2022
(SageMath)
CF = ComplexBallField(100)  # computes a(0)= 1!
def a(n): return Integer(CF(-2).hypergeometric([1, -n], [2-n], regularized=True))
print([a(n) for n in range(23)]) # Peter Luschny, Apr 26 2024

Crossrefs

Cf. A000079, A000142.

Sequence in context: A020530 A337061 A147998 * A054640 A139486 A260622

Adjacent sequences: A052673 A052674 A052675 * A052677 A052678 A052679

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved