A055217 a(n) = sum of the first n coefficients of (1+x+x^2)^n.
1, 3, 10, 31, 96, 294, 897, 2727, 8272, 25048, 75747, 228826, 690691, 2083371, 6280650, 18925047, 57002616, 171633840, 516632307, 1554702516, 4677501237, 14069962041, 42314975352, 127240600050, 382555886571, 1150026301089
Offset: 0
Keywords
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, The Combinatorics of Motzkin Polyominoes, arXiv:2401.06228 [math.CO], 2024.
- Taras Goy and Mark Shattuck, Determinant identities for the Catalan, Motzkin and Schröder numbers, Art Disc. Appl. Math., Vol. 7, No. 1 (2024).
Programs
-
Haskell
a055217 n = sum $ take (n + 1) $ a027907_row (n + 1) -- Reinhard Zumkeller, Jan 22 2013
-
Maple
a := n -> simplify((3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2): seq(a(n), n=0..25); # Peter Luschny, May 12 2016
-
Mathematica
Total/@Table[Take[CoefficientList[Expand[(1+x+x^2)^n],x],n],{n,30}] (* Harvey P. Dale, Aug 14 2011 *) -
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 */
-
PARI
a(n) = my(v=Vec((1+'x+'x^2)^n)); sum(k=1, n, v[k]);
Formula
From Paul Barry, Jan 20 2008: (Start)
Binomial transform of A117186.
G.f.: (1+x-sqrt(1-2x-3x^2))/(2x*(1-2x-3x^2)).
a(n) = (3^(n+1) + A002426(n+1))/2. (End)
From Vladimir Kruchinin, Aug 11 2010: (Start)
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).
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)
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
a(n) = 3^n * 3/2 + O(3^n/sqrt(n)). - Charles R Greathouse IV, Dec 02 2015
From Peter Luschny, May 12 2016: (Start)
a(n) = (3^(n+1) - hypergeom([-(n+1)/2, -n/2], [1], 4))/2.
a(n) = (3^(n+1) - GegenbauerC(n+1,-n-1,-1/2))/2. (End)
Extensions
New description from Paul D. Hanna, Oct 09 2003