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A144297 BINOMIAL transform of A001515.

%I A144297 #15 Sep 28 2023 14:24:08
%S A144297 1,3,12,65,465,4212,46441,604389,9071250,154267865,2931801639,
%T A144297 61578273462,1416474723373,35415138314415,956276678789100,
%U A144297 27733572777976973,859779201497486829,28373745267763162716,993110842735800666085,36746019445535955976665
%N A144297 BINOMIAL transform of A001515.
%H A144297 Vincenzo Librandi, <a href="/A144297/b144297.txt">Table of n, a(n) for n = 0..200</a>
%H A144297 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A144297 From _Vaclav Kotesovec_, Oct 20 2012: (Start)
%F A144297 E.g.f.: exp(1+x-sqrt(1-2*x))/sqrt(1-2*x).
%F A144297 Recurrence: a(n) = (2*n+1)*a(n-1) - (4*n-5)*a(n-2) + 2*(n-2)*a(n-3).
%F A144297 a(n) ~ 2^(n+1/2)*n^n/exp(n-3/2). (End)
%F A144297 a(n) = Sum_{j=0..n} binomial(n,j)*A001515(j). - _G. C. Greubel_, Sep 28 2023
%t A144297 CoefficientList[Series[E^(1+x-Sqrt[1-2*x])/Sqrt[1-2*x],{x,0,20}],x]*Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A144297 (Magma) I:=[1,3,12]; [n le 3 select I[n] else (2*n-1)*Self(n-1) -(4*n-9)*Self(n-2) +2*(n-3)*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Sep 28 2023
%o A144297 (SageMath)
%o A144297 def A144297_list(prec):
%o A144297     P.<x> = PowerSeriesRing(QQ, prec)
%o A144297     return P( exp(1+x-sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
%o A144297 A144297_list(40) # _G. C. Greubel_, Sep 28 2023
%Y A144297 Cf. A001515.
%K A144297 nonn
%O A144297 0,2
%A A144297 _N. J. A. Sloane_, Dec 04 2008