A124431 - OEIS

%I #40 Dec 10 2024 05:27:29

%S 1,3,9,29,97,331,1145,4001,14089,49915,177713,635293,2278841,8198227,

%T 29567729,106872961,387038993,1404052659,5101219929,18559193245,

%U 67605310097,246541193883,899999057385,3288522934433,12026324883865

%N a(n) = Sum_{k=0..n} 2^k*C([(n+k)/2],k)*C([(n+k+1)/2],k) where [x]=floor(x).

%C This is the inverse Motzkin transform of A026378 assuming offset 1 here. - _R. J. Mathar_, Jul 07 2009

%C Hankel transform is Somos-4 variant A162547. - _Paul Barry_, Jan 09 2011

%C 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

%C 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

%H G. C. Greubel, <a href="/A124431/b124431.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} 2^k*A124428(n+k,k).

%F 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

%F 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

%F 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

%F a(n) ~ sqrt(14*sqrt(3)-24) * (2+sqrt(3))^(n+2) / (2*sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Feb 03 2014

%F 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

%F 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

%e G.f. = 1 + 3*x + 9*x^2 + 29*x^3 + 97*x^4 + 331*x^5 + 1145*x^6 + 4001*x^7 + ...

%t 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 *)

%t CoefficientList[Series[(Sqrt[(1+x^2)/(1-4*x+x^2)] -1)/(2*x), {x,0,30}],x] (* _G. C. Greubel_, Feb 26 2019 *)

%o (PARI) a(n)=sum(k=0,n,2^k*binomial((n+k)\2,k)*binomial((n+k+1)\2,k))

%o (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

%o (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

%o (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

%Y Cf. A124428.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 31 2006