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A124431
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a(n) = Sum_{k=0..n} 2^k*C([(n+k)/2],k)*C([(n+k+1)/2],k) where [x]=floor(x).
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2
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1, 3, 9, 29, 97, 331, 1145, 4001, 14089, 49915, 177713, 635293, 2278841, 8198227, 29567729, 106872961, 387038993, 1404052659, 5101219929, 18559193245, 67605310097, 246541193883, 899999057385, 3288522934433, 12026324883865
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OFFSET
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0,2
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COMMENTS
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This is the inverse Motzkin transform of A026378 assuming offset 1 here. - R. J. Mathar, Jul 07 2009
a(n) is the number of peakless Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=29 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH and 2 paths of shape UHD. - Emeric Deutsch, May 03 2011
Conjecture: (n+1)*a(n) -2*(2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(5-2*n)*a(n-3) +(n-3)*a(n-4) =0. - R. J. Mathar, Aug 09 2012
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} 2^k*A124428(n+k,k).
G.f.: (((x^2+1)*(1-4*x+x^2))^(1/2) - (1-4*x+x^2))/(2*x*(1-4*x+x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
G.f.: (1/(1-4*x+x^2))*c(-x/(1-4*x+x^2)), c(x) the g.f. of A000108. - Paul Barry, Jan 09 2011
G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (1+x^2)*(4*k+2) - x*(1+x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1+x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) ~ sqrt(14*sqrt(3)-24) * (2+sqrt(3))^(n+2) / (2*sqrt(3*Pi*n)). - Vaclav Kotesovec, Feb 03 2014
0 = a(n)*(+a(n+1) - 6*a(n+2) + 6*a(n+3) - 18*a(n+4) + 5*a(n+5)) + a(n+1)*(-2*a(n+1) + 14*a(n+2) - 10*a(n+3) + 61*a(n+4) - 18*a(n+5)) + a(n+2)*(+4*a(n+2) - 28*a(n+3) - 10*a(n+4) + 6*a(n+5)) + a(n+3)*(+4*a(n+3) + 14*a(n+4) - 6*a(n+5)) + a(n+4)*(-2*a(n+4) + a(n+5)) if n>=0. - Michael Somos, Aug 06 2014
Conjecture: +(n+1)*a(n) +2*(-2*n-1)*a(n-1) +2*(n-1)*a(n-2) +2*(-2*n+5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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G.f. = 1 + 3*x + 9*x^2 + 29*x^3 + 97*x^4 + 331*x^5 + 1145*x^6 + 4001*x^7 + ...
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MATHEMATICA
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Table[Sum[2^k Binomial[Floor[(n+k)/2], k]Binomial[Floor[(n+k+1)/2], k], {k, 0, n}], {n, 0, 30}] - Harvey P. Dale, May 20 2012
CoefficientList[Series[(Sqrt[(1+x^2)/(1-4*x+x^2)] -1)/(2*x), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)
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PROG
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(PARI) a(n)=sum(k=0, n, 2^k*binomial((n+k)\2, k)*binomial((n+k+1)\2, k))
(PARI) my(x='x+O('x^30)); Vec((sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x)) \\ G. C. Greubel, Feb 26 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x) )); // G. C. Greubel, Feb 26 2019
(Sage) ((sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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