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A143013 Number of Motzkin n-paths with two kinds of level steps one of which is a final step. 3
1, 2, 3, 7, 17, 43, 114, 310, 861, 2433, 6970, 20198, 59101, 174373, 518179, 1549545, 4659399, 14079553, 42732230, 130208246, 398174723, 1221573603, 3758835953, 11597578995, 35872937745, 111216324015, 345539568900, 1075693015920 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Hankel transform is the (4,-5) Somos-4 variant A171422. - Paul Barry, Dec 08 2009
LINKS
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
FORMULA
Words on alphabet {U,D,L,F} of length n where U is upstep, D is downstep, L and F are level steps and F can only be immediately followed by D or end of word with defining equation A = 1 + F + LA + UADA.
When convolved with itself yields first difference shifted left one place.
G.f. A(x) satisfies: A(x) = 1 + x + A(x)*x + (A(x)*x)^2.
G.f.: (1+x) / (1-x -(x^2 + x^3) / (1-x -(x^2 + x^3) / (1-x -...))).
G.f.: (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3)) / (2*x^2).
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([0,1,2]). - Gary W. Adamson, Oct 27 2008
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(1-n)*a(n-2) +2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Oct 25 2012
a(n) = Sum_{i=0..n+2} Sum_{k=1..n-i+2} C(i-1,k-1)*C(k,n-k-i+2)*C(k+i-2,i-1)/k. - Vladimir Kruchinin, May 06 2018
EXAMPLE
A = 1 + (L + F) + (LL + LF + UD) + (LLL + LLF + LUD + UDL + UDF + ULD + UFD) + ...
G.f. = 1 + 2*x + 3*x^2 + 7*x^3 + 17*x^4 + 43*x^5 + 114*x^6 + 310*x^7 + 861*x^8 + ...
MATHEMATICA
CoefficientList[Series[(1-x -Sqrt[1-2*x-3*x^2-4*x^3])/(2*x^2), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)
PROG
(PARI) {a(n) = if(n<0, 0, polcoeff( (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3 + x^3*O(x^n))) / (2*x^2), n))}
(PARI) x='x+O('x^30); Vec((1-x-(1-2*x-3*x^2-4*x^3)^(1/2))/(2*x^2)) \\ Altug Alkan, May 06 2018
(Maxima)
a(n):=sum(sum((binomial(i-1, k-1)*binomial(k, n-k-i+2)*binomial(k+i-2, i-1))/k, k, 1, n-i+2), i, 0, n+2); /* Vladimir Kruchinin, May 06 2018 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x -Sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2) )); // G. C. Greubel, Feb 26 2019
(Sage) ((1-x -sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019
CROSSREFS
Sequence in context: A178178 A333499 A257553 * A113483 A173868 A274291
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 15 2008
STATUS
approved

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Last modified February 21 02:30 EST 2024. Contains 370219 sequences. (Running on oeis4.)