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A006605
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Number of modes of connections of 2n points.
(Formerly M2899)
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14
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1, 1, 3, 11, 46, 207, 979, 4797, 24138, 123998, 647615, 3428493, 18356714, 99229015, 540807165, 2968468275, 16395456762, 91053897066, 508151297602, 2848290555562, 16028132445156, 90516256568235, 512831902620465, 2914112388802779, 16604034506299314
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OFFSET
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0,3
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COMMENTS
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Number of paths of semilength n staying weakly within the 1st quadrant starting at (0,0) and ending on the X-axis using steps (1,1), (1,-1) and (1,3). - David Scambler, Jun 21 2013
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
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LINKS
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FORMULA
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Reference gives explicit formula.
G.f.: A(x) = (1/x)*serreverse(x/G(x)) where G(x) is g.f. of A001006 (Motzkin numbers). G.f. satisfies: A(x)^2 = (1/x)*serreverse( x/(1+x+x^2)^2 ). - Paul D. Hanna, Mar 20 2005
G.f.: revogf is 1/2*(-x+1+(-(1+x)*(-1+3*x))^(1/2))*x. - Simon Plouffe, Master's Thesis, UQAM 1992
a(n) = (1/(2*n+1))*Sum_{j=0...2*n+1} binomial(j,2*j-2-3*n)*binomial(2*n+1,j). - Vladimir Kruchinin, Dec 24 2010
a(n) ~ sqrt(89 + 277/sqrt(13)) * ((70 + 26*sqrt(13))/27)^n / (9*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 15 2013
With interpolated zeros, the o.g.f. = series reversion of x/(1 + x^2 + x^4) = x + x^3 + 3*x^5 + 11*x^7 + 46*x^9 + .... - Peter Bala, Dec 17 2013
Conjecture: 3*n*(3*n+2)*(3*n+1)*a(n) + (-275*n^3 + 475*n^2 - 328*n + 68)*a(n-1) + 2*(2*n-3)*(139*n^2 - 507*n + 398)*a(n-2) + 180*(2*n-5)*(n-2)*(2*n-3)*a(n-3) = 0. - R. J. Mathar, May 30 2014
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MAPLE
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series(RootOf(x^2*g^4+x*g^2-g+1, g), x=0, 20); # Mark van Hoeij, Nov 16 2011
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1,
2*((910*n^4 -1085*n^3 +227*n^2 +92*n -24) *a(n-1)
+(936*n^4 -2520*n^3 +1710*n^2 +90*n-216) *a(n-2))/
(3*n *(117*n^3 +36*n^2 -55*n -18)))
end:
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MATHEMATICA
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PROG
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(PARI) {a(n)=polcoeff(((1/x)*serreverse(x/(1+x+x^2)^2+x^2*O(x^n)))^(1/2), n)} \\ Paul D. Hanna
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 20 2005
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STATUS
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approved
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