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%I #21 Sep 28 2023 15:38:19
%S 1,4,25,199,1936,22411,301939,4649800,80654599,1556992441,33120019516,
%T 769887934729,19419368959225,528311452144204,15421347559288441,
%U 480784227676809991,15945180393017896024,560549114426134288675
%N Second differences of A001515 (or A144301).
%H G. C. Greubel, <a href="/A144647/b144647.txt">Table of n, a(n) for n = 0..400</a>
%F G.f.: (1-x)^2/(x*Q(0)) + 1 - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 18 2013
%F G.f.: T(0)*(1-x)/x + 1 - 1/x, where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 26 2013
%F From _G. C. Greubel_, Sep 28 2023: (Start)
%F a(n) = A001515(n+1) - 2*A001515(n) + A001515(n).
%F a(n) = 2*A001515(n+1) - (2*n+3)*A001515(n).
%F a(n) = ( ((2*n-1)*(4*n^2 - 4*n + 5))*a(n-1) + (4*n^2 + 3)*a(n-2) )/(4*n^2 - 8*n + 7), with a(0) = 1, a(1) = 4.
%F E.g.f.: (-1 + 4*x + 2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3. (End)
%p A001515 := proc(n) simplify(hypergeom([n+1,-n],[],-1/2)) ; end: A144647 := proc(n) if n =0 then A001515(n) ; else A001515(n+1)-2*A001515(n)+A001515(n-1) ; fi; end: seq(A144647(n),n=0..30) ; # _R. J. Mathar_, Feb 01 2009
%t Join[{1},Differences[RecurrenceTable[{a[0]==1,a[1]==2,a[n]== (2n-1)a[n-1]+ a[n-2]},a[n],{n,25}],2]] (* _Harvey P. Dale_, Jun 18 2011 *)
%o (Magma) [n le 2 select 4^(n-1) else ( ((2*n-3)*(4*n^2-12*n+13))*Self(n-1) + (4*n^2-8*n+7)*Self(n-2) )/(4*n^2-16*n+19): n in [1..30]]; // _G. C. Greubel_, Sep 28 2023
%o (SageMath)
%o def A144647_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( (-1+4*x+2*(1-x)*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(sqrt(1-2*x))^3 ).egf_to_ogf().list()
%o A144647_list(40) # _G. C. Greubel_, Sep 28 2023
%Y Cf. A001515, A144301, A144498.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 26 2009
%E More terms from _R. J. Mathar_, Feb 01 2009
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