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A055217
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a(n) = sum of the first n coefficients of (1+x+x^2)^n.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| 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; - Paul Barry (pbarry(AT)wit.ie), Jan 20 2008
Logarithm g.f. log(1/(1-M(x))=sum(n>0, a(n)/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. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 11 2010]
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
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MATHEMATICA
| Total/@Table[Take[CoefficientList[Expand[(1+x+x^2)^n], x], n], {n, 30}] (* From Harvey P. Dale, Aug 14 2011 *)
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PROG
| (Other) a(n):=sum(sum(binomial(n, j)*binomial(j, 2*j-n-k), j, ceiling((n+k)/2), n), k, 1, n); (for Maxima) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 11 2010]
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CROSSREFS
| T(2n+1, n), array T as in A055216.
Sequence in context: A180432 A192337 A106517 * A097472 A068094 A100058
Adjacent sequences: A055214 A055215 A055216 * A055218 A055219 A055220
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), May 07 2000
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EXTENSIONS
| New description from Paul D. Hanna (pauldhanna(AT)juno.com), Oct 09 2003
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