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

%I A098558 #52 Feb 16 2025 08:32:54
%S A098558 1,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,
%T A098558 12454041600,174356582400,2615348736000,41845579776000,
%U A098558 711374856192000,12804747411456000,243290200817664000,4865804016353280000,102181884343418880000,2248001455555215360000
%N A098558 Expansion of e.g.f. (1+x)/(1-x).
%C A098558 Essentially the same as A052849: a(0)=0 and a(n) = A052849(n) for n>=1.
%C A098558 Equals row sums (unsigned) of triangle A158471. - _Gary W. Adamson_, Mar 20 2009
%C A098558 Also the number of graceful labelings of the star graph on n+1 nodes. - _Eric W. Weisstein_, Mar 31 2020
%H A098558 G. C. Greubel, <a href="/A098558/b098558.txt">Table of n, a(n) for n = 0..445</a>
%H A098558 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>
%H A098558 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarGraph.html">Star Graph</a>
%F A098558 a(n) = 2*n! - 0^n.
%F A098558 a(n) = Sum_{k=0..n} (k+1) * A008290(n,k). - _Alois P. Heinz_, Mar 11 2022
%F A098558 Sum_{n>=0} 1/a(n) = (e+1)/2. - _Amiram Eldar_, Feb 02 2023
%F A098558 a(n) = HypergeomRegularized([1, -n], [2 - n], -1). - _Peter Luschny_, Apr 26 2024
%t A098558 Join[{1}, 2*Range[30]!] (* _G. C. Greubel_, Jan 17 2018 *)
%t A098558 With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 05 2021 *)
%t A098558 a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];
%t A098558 Table[a[n], {n, 0, 22}]  (* _Peter Luschny_, Apr 26 2024 *)
%o A098558 (PARI) concat([1], vector(30, n, 2*n!)) \\ _G. C. Greubel_, Jan 17 2018
%o A098558 (Magma) [1] cat [2*Factorial(n): n in [1..30]]; // _G. C. Greubel_, Jan 17 2018
%o A098558 (SageMath)
%o A098558 CF = ComplexBallField(100)
%o A098558 def a(n):
%o A098558     return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))
%o A098558 print([a(n) for n in range(23)]) # _Peter Luschny_, Apr 26 2024
%Y A098558 Cf. A000007, A000142, A008290.
%Y A098558 Row sums of A008518 and of A128564.
%Y A098558 Cf. A158471.
%K A098558 easy,nonn
%O A098558 0,2
%A A098558 _Paul Barry_, Sep 14 2004