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%I #47 Jan 15 2024 09:36:46
%S 1,3,10,31,96,294,897,2727,8272,25048,75747,228826,690691,2083371,
%T 6280650,18925047,57002616,171633840,516632307,1554702516,4677501237,
%U 14069962041,42314975352,127240600050,382555886571,1150026301089
%N a(n) = sum of the first n coefficients of (1+x+x^2)^n.
%H Reinhard Zumkeller, <a href="/A055217/b055217.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 From _Paul Barry_, Jan 20 2008: (Start)
%F Binomial transform of A117186.
%F G.f.: (1+x-sqrt(1-2x-3x^2))/(2x*(1-2x-3x^2)).
%F a(n) = (3^(n+1) + A002426(n+1))/2. (End)
%F From _Vladimir Kruchinin_, Aug 11 2010: (Start)
%F Logarithm g.f.: log(1/(1-M(x)) = Sum_{n>0} a(n)/n*x^n, M(x) - o.g.f Motzkin numbers (A001006).
%F 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 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 a(n) = 3^n * 3/2 + O(3^n/sqrt(n)). - _Charles R Greathouse IV_, Dec 02 2015
%F From _Peter Luschny_, May 12 2016: (Start)
%F a(n) = (3^(n+1) - hypergeom([-(n+1)/2, -n/2], [1], 4))/2.
%F a(n) = (3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2. (End)
%p a := n -> simplify((3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2):
%p seq(a(n), n=0..25); # _Peter Luschny_, May 12 2016
%t Total/@Table[Take[CoefficientList[Expand[(1+x+x^2)^n],x],n],{n,30}] (* _Harvey P. Dale_, Aug 14 2011 *)
%o (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 (Haskell)
%o a055217 n = sum $ take (n + 1) $ a027907_row (n + 1)
%o -- _Reinhard Zumkeller_, Jan 22 2013
%o (PARI) a(n) = my(v=Vec((1+'x+'x^2)^n)); sum(k=1, n, v[k]);
%Y T(2n+1, n), array T as in A055216.
%Y Cf. A000244, A002426, A027914.
%Y Cf. A001006, A117186.
%K nonn
%O 0,2
%A _Clark Kimberling_, May 07 2000
%E New description from _Paul D. Hanna_, Oct 09 2003
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