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A253918 Number of Motzkin n-paths with two kinds of level steps both of which are final steps.

%I A253918 #14 Jun 03 2017 19:00:58
%S A253918 1,2,1,4,6,12,29,56,134,300,682,1624,3772,9016,21597,51888,126086,
%T A253918 306636,750398,1843864,4543604,11242088,27887730,69378192,173050396,
%U A253918 432596216,1083862132,2720961520,6843557944,17242789104,43514507997,109983815264,278385212358
%N A253918 Number of Motzkin n-paths with two kinds of level steps both of which are final steps.
%C A253918 Number of words of length n on alphabet {U,D,L,R} where U is upstep, D is downstep, L and R are level steps and can only be immediately followed by D or end of word. Defining equation is A = 1 + L + R + UADA.
%H A253918 G. C. Greubel, <a href="/A253918/b253918.txt">Table of n, a(n) for n = 0..1000</a>
%F A253918 When convolved with itself yields sequence shifted left two places.
%F A253918 G.f. A(x) satisfies A(x) = 1 + 2*x + (A(x)*x)^2.
%F A253918 G.f.: (1 - sqrt(1 - 4*x^2 - 8*x^3)) / (2*x^2).
%F A253918 0 = a(n)*(8*n+4) + a(n+1)*(4*n+8) + a(n+3)*(-n-5) for n>=-1.
%F A253918 a(n) = Sum_{k=0..n}((binomial(k+1,n-2*k)*binomial(2*k,k)*2^(n-2*k))/(k+1)). - _Vladimir Kruchinin_, Mar 11 2016
%F A253918 a(n) ~ sqrt(3-4*c^2) * 4^n * c^n * (1+2*c)^(n+1) / (sqrt(Pi)*n^(3/2)), where c = 0.37743883312334638... is the root of the equation 4*c^2*(1 + 2*c) = 1. - _Vaclav Kotesovec_, Mar 10 2016
%e A253918 G.f. = 1 + 2*x + x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 29*x^6 + 56*x^7 + ...
%e A253918 A = 1 + (L + R) + (UD) + (ULD + URD + UDR + UDL) + (UDUD + UUDD + ULDL + ULDR + URDL + URDR) + ...
%t A253918 a[ n_] := SeriesCoefficient[ (1 - Sqrt[1 - 4 x^2 - 8x^3]) / (2 x^2), {x, 0, n}];
%o A253918 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - sqrt(1 - 4*x^2 - 8*x^3  + x^3*O(x^n))) / (2*x^2), n))};
%o A253918 (PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=0, n\2, A = 1 + 2*x + sqr(x*A)); polcoeff( A, n))};
%o A253918 (Maxima)
%o A253918 a(n):=sum((binomial(k+1,n-2*k)*binomial(2*k,k)*2^(n-2*k))/(k+1),k,0,n); /* _Vladimir Kruchinin_, Mar 11 2016 */
%Y A253918 Cf. A143013.
%K A253918 nonn
%O A253918 0,2
%A A253918 _Michael Somos_, Jan 19 2015