%I #52 Feb 16 2025 08:32:54

%S 1,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,

%T 12454041600,174356582400,2615348736000,41845579776000,

%U 711374856192000,12804747411456000,243290200817664000,4865804016353280000,102181884343418880000,2248001455555215360000

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

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

%C Equals row sums (unsigned) of triangle A158471. - _Gary W. Adamson_, Mar 20 2009

%C Also the number of graceful labelings of the star graph on n+1 nodes. - _Eric W. Weisstein_, Mar 31 2020

%H G. C. Greubel, <a href="/A098558/b098558.txt">Table of n, a(n) for n = 0..445</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarGraph.html">Star Graph</a>

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

%F a(n) = Sum_{k=0..n} (k+1) * A008290(n,k). - _Alois P. Heinz_, Mar 11 2022

%F Sum_{n>=0} 1/a(n) = (e+1)/2. - _Amiram Eldar_, Feb 02 2023

%F a(n) = HypergeomRegularized([1, -n], [2 - n], -1). - _Peter Luschny_, Apr 26 2024

%t Join[{1}, 2*Range[30]!] (* _G. C. Greubel_, Jan 17 2018 *)

%t With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 05 2021 *)

%t a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];

%t Table[a[n], {n, 0, 22}] (* _Peter Luschny_, Apr 26 2024 *)

%o (PARI) concat([1], vector(30, n, 2*n!)) \\ _G. C. Greubel_, Jan 17 2018

%o (Magma) [1] cat [2*Factorial(n): n in [1..30]]; // _G. C. Greubel_, Jan 17 2018

%o (SageMath)

%o CF = ComplexBallField(100)

%o def a(n):

%o return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))

%o print([a(n) for n in range(23)]) # _Peter Luschny_, Apr 26 2024

%Y Cf. A000007, A000142, A008290.

%Y Row sums of A008518 and of A128564.

%Y Cf. A158471.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 14 2004