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Number of 2-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.
1

%I #28 Sep 24 2016 18:24:43

%S 1,3,10,37,152,690,3422,18257,103144,608730,3713524,23235490,

%T 148281656,961255200,6311395814,41878914665,280365966232,

%U 1891270498050,12842102343820,87703053156406,601999871121200,4150859861430252,28736613316786220,199671324115916570

%N Number of 2-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.

%F G.f.: 6/(5-14*x+sqrt(1-8*x+4*x^2))=1/(1-2*x-x*F(x)), where F(x) is the g.f. of the sequence A007564.

%F a(n) ~ sqrt(1275+746*sqrt(3)) * (2*(2+sqrt(3)))^n / (121*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015

%F a(n) = Sum_{k=0..n}((k+1)*(Sum_{j=0..k}(2^j*(-1)^(-k+j)*binomial(k-j,j)))*Sum_{j=0..n+1}(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j)))/(n+1). - _Vladimir Kruchinin_, Mar 08 2016

%F Conjecture: 2*n*a(n) +3*(-9*n+8)*a(n-1) +4*(28*n-39)*a(n-2) +4*(-43*n+81)*a(n-3) +64*(n-3)*a(n-4)=0. - _R. J. Mathar_, Sep 24 2016

%e a(2) = 10 because we have H1H1, H1H2, H2H1, H2H2, UDH1, UDH2, H1UD, H2UD, UDUD and UUDD.

%t CoefficientList[Series[1/(1 - 2 x - x (1 + 2 x - Sqrt[1 - 8 x + 4 x^2])/(6 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Apr 21 2015 *)

%o (Maxima)

%o a(n):=sum((k+1)*(sum(2^j*(-1)^(-k+j)*binomial(k-j,j),j,0,k))*sum(binomial(j,-n-k+2*j-2)*4^(-n-k+2*j-2)*3^(n-j+1)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Mar 08 2016 */

%Y Cf. A007564, A224071.

%K nonn

%O 0,2

%A _José Luis Ramírez Ramírez_, Apr 20 2015

%E More terms from _Vincenzo Librandi_, Apr 21 2015