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A143013
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Number of Motzkin n-paths with two kinds of level steps one of which is a final step.
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1
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Hankel transform is the (4,-5) Somos-4 variant A171422. [From Paul Barry (pbarry(AT)wit.ie), Dec 08 2009]
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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).
Comment from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 27 2008: 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]).
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EXAMPLE
| A = 1 + (L + F) + (LL + LF + UD) + (LLL + LLF + LUD + UDL + UDF + ULD + UFD) + ...
1 + 2*x + 3*x^2 + 7*x^3 + 17*x^4 + 43*x^5 + 114*x^6 + 310*x^7 + 861*x^8 + ...
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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))}
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CROSSREFS
| Cf. cf. A007477.
Sequence in context: A135364 A051291 A178178 * A113483 A173868 A176074
Adjacent sequences: A143010 A143011 A143012 * A143014 A143015 A143016
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jul 15 2008
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