A098558 Expansion of e.g.f. (1+x)/(1-x).

1, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

Author

Paul Barry, Sep 14 2004

Keywords

• easy
• nonn

Comments

Essentially the same as A052849: a(0)=0 and a(n) = A052849(n) for n>=1.

Equals row sums (unsigned) of triangle A158471. - Gary W. Adamson, Mar 20 2009

Also the number of graceful labelings of the star graph on n+1 nodes. - Eric W. Weisstein, Mar 31 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• Eric Weisstein's World of Mathematics, Graceful Labeling
• Eric Weisstein's World of Mathematics, Star Graph

Crossrefs

Cf. A000007, A000142, A008290.

Row sums of A008518 and of A128564.

Cf. A158471.

Programs

Magma

[1] cat [2*Factorial(n): n in [1..30]]; // G. C. Greubel, Jan 17 2018

Mathematica

Join[{1}, 2*Range[30]!] (* G. C. Greubel, Jan 17 2018 *)
With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2021 *)
a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 26 2024 *)

PARI

concat([1], vector(30, n, 2*n!)) \\ G. C. Greubel, Jan 17 2018

SageMath

CF = ComplexBallField(100)
def a(n):
    return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))
print([a(n) for n in range(23)]) # Peter Luschny, Apr 26 2024

Formula

a(n) = 2*n! - 0^n.

a(n) = Sum_{k=0..n} (k+1) * A008290(n,k). - Alois P. Heinz, Mar 11 2022

Sum_{n>=0} 1/a(n) = (e+1)/2. - Amiram Eldar, Feb 02 2023

a(n) = HypergeomRegularized([1, -n], [2 - n], -1). - Peter Luschny, Apr 26 2024