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%I #26 Jun 12 2021 02:27:03
%S 0,1,5,24,116,568,2820,14184,72180,371112,1925380,10068728,53023860,
%T 280969560,1497072132,8016213960,43114424308,232817773640,
%U 1261793848836,6861179441880,37421756333172,204671007577464,1122275850740996,6168352091629864,33977333521770996,187539324760522728
%N Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).
%C a(n) = Sum_{k=0..n} k*A110189(n,k).
%H Vincenzo Librandi, <a href="/A110190/b110190.txt">Table of n, a(n) for n = 0..200</a>
%H Luis Verde-Star <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%F G.f.: x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2, where R = 1+x*R+x*R^2 = (1-x-sqrt(1-6*x+x^2))/(2*x) is the g.f. for the large Schroeder numbers (A006318).
%F Recurrence: (n+2)*(n+3)*a(n) = (5*n^2+29*n+10)*a(n-1) + (5*n^2-59*n+142)*a(n-2) - (n-6)*(n-5)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 3*2^(1/4)*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%F G.f. A(x) satisfies x^2*A(x)^2 = (x^4 - 7*x^3 + 12*x^2 - 7*x + 1)*A(x) + (-x^3 + 2*x^2 - x). - _Joerg Arndt_, May 16 2013
%F a(n) = Sum_{k=0..n} ((k+1)*Sum_{i=0..n-k} (binomial(n+1,n-k-i)*binomial(n+i,n))/ (n+1)*a113127(k)). - _Vladimir Kruchinin_, Mar 13 2016
%e a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.
%p R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..26);
%t CoefficientList[Series[x*(1-x-2*x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+2*x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x))^2)/(1-3*x-x*((1-x-Sqrt[1-6*x+x^2])/(2*x))+x^2+x^2*((1-x-Sqrt[1-6*x+x^2])/(2*x)))^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o (PARI)
%o x = 'x+O('x^66);
%o R = (1-x-sqrt(1-6*x+x^2))/(2*x);
%o gf = x*(1-x-2*x*R+x^2+2*x^2*R+x^2*R^2)/(1-3*x-x*R+x^2+x^2*R)^2;
%o concat([0],Vec(gf))
%o \\ _Joerg Arndt_, May 16 2013
%o (Maxima)
%o a113127(n):=if n=0 then 1 else if n=1 then 3 else 4*n-2;
%o a(n):=sum((k+1)*sum(binomial(n+1,n-k-i)*binomial(n+i,n),i,0,n-k)/(n+1)*a113127(k),k,0,n); /* _Vladimir Kruchinin_, Mar 13 2016 */
%Y Cf. A006318, A110189.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jul 15 2005
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