A250307 Number of 2-colored Schroeder paths of semilength n in which there are no (2,0)-steps at level 1.
1, 3, 10, 37, 152, 690, 3422, 18257, 103144, 608730, 3713524, 23235490, 148281656, 961255200, 6311395814, 41878914665, 280365966232, 1891270498050, 12842102343820, 87703053156406, 601999871121200, 4150859861430252, 28736613316786220, 199671324115916570
Offset: 0
Keywords
Examples
a(2) = 10 because we have H1H1, H1H2, H2H1, H2H2, UDH1, UDH2, H1UD, H2UD, UDUD and UUDD.
Programs
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Mathematica
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 *) -
Maxima
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 */
Formula
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.
a(n) ~ sqrt(1275+746*sqrt(3)) * (2*(2+sqrt(3)))^n / (121*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
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
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
Extensions
More terms from Vincenzo Librandi, Apr 21 2015