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1, 4, 18, 86, 426, 2162, 11166, 58438, 309042, 1648154, 8851206, 47813790, 259585002, 1415431266, 7747200558, 42545600310, 234346445154, 1294260644906, 7165245015510, 39754745775886, 221009855334426, 1230909476804594, 6867024985408638, 38369226561522086
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OFFSET
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0,2
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COMMENTS
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a(n) is also the number of Schröder paths of semilength n (paths from (0, 0) to (2*n, 0), using only single steps northeast or southeast (steps (1, 1) or (1, -1)) or double steps east (steps (2, 0)), that never fall below the x-axis) in which the (2,0)-steps that are on the horizontal axis come in 3 colors (see Oste and Van der Jeugt, Section 7).
Example: a(2) = 18 because from the origin to the point (4,0) we have 3^2 = 9 paths of type HH, 3 paths of type HUD, 3 paths of type UDH as well as the paths UDUD, UUDD, and UHD.
It follows that the sequence may be calculated as the leading diagonal of the lower triangular array (T(n,k))n,k>=0 defined by the relations: T(n,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-1) for 1 <= k <= n-1 and T(n,n) = 3*T(n-1,n-1) + T(n,n-1). The array begins: [1], [1, 4], [1, 6, 18], [1, 8, 32, 86], [1, 10, 50, 168, 426]. (End)
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LINKS
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FORMULA
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G.f.: (-1 + 5*x + sqrt(1 - 6*x + x^2)) / (2 * (x - 6*x^2)) = 2 / (1 - 5*x + sqrt(1 - 6*x + x^2)).
G.f.: A(x) = 1 / (1 - 5*x + (x - 6*x^2) * A(x)) = 1 + x * A(x) * (5 - A(x) * (1 - 6*x)).
HANKEL transform is A006125. HANKEL transform with 1 prepended is A127850(n+1).
Conjecture: (n+1)*a(n) +3*(-4*n-1)*a(n-1) +(37*n-20)*a(n-2) +6*(-n+2)*a(n-3)=0. - R. J. Mathar, May 23 2014
a(n) = Sum_{k=0..n}((k+1)*Sum_{j=0..n+1}(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, Mar 13 2016
a(n) ~ (1+sqrt(2))^(2*n+5) / (2^(3/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 13 2016
G.f.: 1/(1-3*x -x/(1-x -x/(1-x -x/(1-x - ... )))) (continued fraction) = 1/(1 - 3*x - x*S(x)), where S(x) is the generating function of the large Schröder numbers A001003. - Peter Bala, Apr 23 2017
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EXAMPLE
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1 + 4*x + 18*x^2 + 86*x^3 + 426*x^4 + 2162*x^5 + 11166*x^6 + 58438*x^7 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 2 / (1 - 5 x + Sqrt[1 - 6 x + x^2]), {x, 0, n}]
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 - 5*x + sqrt(1 - 6*x + x^2 + x * O(x^n))), n))}
(Maxima)
a(n):=sum((k+1)*sum(binomial(j, n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1, j), j, 0, n+1), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 13 2016 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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