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A055217 a(n) = sum of the first n coefficients of (1+x+x^2)^n.

%I A055217 #54 Jan 05 2025 09:23:40
%S A055217 1,3,10,31,96,294,897,2727,8272,25048,75747,228826,690691,2083371,
%T A055217 6280650,18925047,57002616,171633840,516632307,1554702516,4677501237,
%U A055217 14069962041,42314975352,127240600050,382555886571,1150026301089
%N A055217 a(n) = sum of the first n coefficients of (1+x+x^2)^n.
%H A055217 Reinhard Zumkeller, <a href="/A055217/b055217.txt">Table of n, a(n) for n = 0..1000</a>
%H A055217 Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, <a href="https://arxiv.org/abs/2401.06228">The Combinatorics of Motzkin Polyominoes</a>, arXiv:2401.06228 [math.CO], 2024.
%H A055217 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1645.d36">Determinant identities for the Catalan, Motzkin and Schröder numbers</a>, Art Disc. Appl. Math., Vol. 7, No. 1 (2024).
%F A055217 From _Paul Barry_, Jan 20 2008: (Start)
%F A055217 Binomial transform of A117186.
%F A055217 G.f.: (1+x-sqrt(1-2x-3x^2))/(2x*(1-2x-3x^2)).
%F A055217 a(n) = (3^(n+1) + A002426(n+1))/2. (End)
%F A055217 From _Vladimir Kruchinin_, Aug 11 2010: (Start)
%F A055217 Logarithm g.f.: log(1/(1-x*M(x))) = Sum_{n>0} a(n-1)/n*x^n, M(x) - o.g.f Motzkin numbers (A001006).
%F A055217 a(n) = sum(sum(binomial(n,j)*binomial(j,2*j-n-k),j,ceiling((n+k)/2),n),k,1,n), n>0. (End)
%F A055217 Conjecture: (n+1)*a(n) -(5*n+1)*a(n-1) +3*(n-1)*a(n-2) +9*(n-1)*a(n-3)=0. - _R. J. Mathar_, Nov 14 2011
%F A055217 a(n) = 3^n * 3/2 + O(3^n/sqrt(n)). - _Charles R Greathouse IV_, Dec 02 2015
%F A055217 From _Peter Luschny_, May 12 2016: (Start)
%F A055217 a(n) = (3^(n+1) - hypergeom([-(n+1)/2, -n/2], [1], 4))/2.
%F A055217 a(n) = (3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2. (End)
%p A055217 a := n -> simplify((3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2):
%p A055217 seq(a(n), n=0..25); # _Peter Luschny_, May 12 2016
%t A055217 Total/@Table[Take[CoefficientList[Expand[(1+x+x^2)^n],x],n],{n,30}] (* _Harvey P. Dale_, Aug 14 2011 *)
%o A055217 (Maxima) a(n):=sum(sum(binomial(n,j)*binomial(j,2*j-n-k),j,ceiling((n+k)/2),n),k,1,n); /* _Vladimir Kruchinin_, Aug 11 2010 */
%o A055217 (Haskell)
%o A055217 a055217 n = sum $ take (n + 1) $ a027907_row (n + 1)
%o A055217 -- _Reinhard Zumkeller_, Jan 22 2013
%o A055217 (PARI) a(n) = my(v=Vec((1+'x+'x^2)^n)); sum(k=1, n, v[k]);
%Y A055217 T(2n+1, n), array T as in A055216.
%Y A055217 Cf. A000244, A002426, A027914.
%Y A055217 Cf. A001006, A117186.
%K A055217 nonn
%O A055217 0,2
%A A055217 _Clark Kimberling_, May 07 2000
%E A055217 New description from _Paul D. Hanna_, Oct 09 2003