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A137256 Binomial transform of 2^n, 2^n, 2^n.

%I A137256 #10 Jun 17 2024 22:45:38
%S A137256 1,2,4,9,21,48,108,243,549,1242,2808,6345,14337,32400,73224,165483,
%T A137256 373977,845154,1909980,4316409,9754749,22044960,49819860,112588947,
%U A137256 254442141,575019162,1299497904,2936762649,6636851721,14998760928
%N A137256 Binomial transform of 2^n, 2^n, 2^n.
%C A137256 Sequence is identical to half its third differences.
%D A137256 Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.
%H A137256 G. C. Greubel, <a href="/A137256/b137256.txt">Table of n, a(n) for n = 0..1000</a>
%H A137256 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3).
%F A137256 a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3).
%F A137256 O.g.f.: (1 -x +x^2)/(1 -3*x +3*x^2 -3*x^3). - _R. J. Mathar_, Apr 02 2008
%p A137256 m:=30; S:=series( (1-x+x^2)/(1-3*x+3*x^2-3*x^3), x, m+1):
%p A137256 seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Apr 10 2021
%t A137256 LinearRecurrence[{3, -3, 3},{1, 2, 4},30] (* _Ray Chandler_, Sep 23 2015 *)
%o A137256 (Magma) I:=[1,2,4]; [n le 3 select I[n] else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..30]]; // _G. C. Greubel_, Apr 10 2021
%o A137256 (Sage)
%o A137256 def A137256_list(prec):
%o A137256     P.<x> = PowerSeriesRing(ZZ, prec)
%o A137256     return P( (1-x+x^2)/(1-3*x+3*x^2-3*x^3) ).list()
%o A137256 A137256_list(30) # _G. C. Greubel_, Apr 10 2021
%Y A137256 Cf. A051164, A052103.
%K A137256 nonn
%O A137256 0,2
%A A137256 _Paul Curtz_, Mar 11 2008
%E A137256 More terms from _R. J. Mathar_, Apr 02 2008