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A225887 a(n) = A212205(2*n + 1). 2
1, 4, 18, 86, 426, 2162, 11166, 58438, 309042, 1648154, 8851206, 47813790, 259585002, 1415431266, 7747200558, 42545600310, 234346445154, 1294260644906, 7165245015510, 39754745775886, 221009855334426, 1230909476804594, 6867024985408638, 38369226561522086 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Peter Bala, Apr 23 2017: (Start)

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)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

R. Oste and J. Van der Jeugt, Motzkin paths, Motzkin polynomials and recurrence relations, Electronic Journal of Combinatorics 22(2) (2015), #P2.8. Section 7

FORMULA

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)).

INVERT transform of A001003(n+1). INVERT transform is A134425.

HANKEL transform is A006125. HANKEL transform with 1 prepended is A127850(n+1).

BINOMIAL transform of A151090.

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

EXAMPLE

1 + 4*x + 18*x^2 + 86*x^3 + 426*x^4 + 2162*x^5 + 11166*x^6 + 58438*x^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 2 / (1 - 5 x + Sqrt[1 - 6 x + x^2]), {x, 0, n}]

PROG

(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 */

CROSSREFS

Cf. A001003, A006125, A127850, A131090, A151090, A212205, A111966.

Sequence in context: A084847 A082685 A111966 * A153294 A164045 A178577

Adjacent sequences:  A225884 A225885 A225886 * A225888 A225889 A225890

KEYWORD

nonn,easy

AUTHOR

Michael Somos, May 19 2013

STATUS

approved

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Last modified October 15 00:04 EDT 2019. Contains 328025 sequences. (Running on oeis4.)