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 A127848 Series reversion of x/(1+6x+5x^2). 3
 0, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Hankel transform is -A127849(n)=-5^C(n,2)*(5^n-1)/4; a(n+1) counts (6,5)-Motzkin paths of length n, where there are 6 colors available for the H(1,0) steps and 5 for the U(1,1) steps. See A078009 for more information. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA G.f.: (1-6x-sqrt(1-12x+16x^2))/(10x); a(n)=sum{k=0..n-1, (1/n)*C(n,k)C(n,k+1)5^k}; a(n+1)=sum{k=0..floor(n/2), C(n, 2k)C(k)6^(n-2k)*5^k}; Recurrence: (n+1)*a(n) = 6*(2*n-1)*a(n-1) - 16*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 20 2012 a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012. Equivalently, a(n) ~ 2^(2*n) * phi^(2*n + 1) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021 a(n) = A078009(n) for n>0. - Philippe Deléham, Apr 03 2013 MATHEMATICA CoefficientList[ InverseSeries[ Series[ x/(1+6x+5x^2), {x, 0, 20}], x], x] (* Jean-François Alcover, May 24 2012 *) CROSSREFS Cf. A078009. Sequence in context: A152107 A143023 A078009 * A113573 A083161 A077147 Adjacent sequences: A127845 A127846 A127847 * A127849 A127850 A127851 KEYWORD easy,nonn AUTHOR Paul Barry, Feb 02 2007 STATUS approved

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Last modified March 1 06:21 EST 2024. Contains 370430 sequences. (Running on oeis4.)