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A035929
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Number of Dyck n-paths starting U^mD^m (an m-pyramid), followed by a pyramid-free Dyck path.
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4
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0, 1, 1, 1, 2, 6, 19, 61, 200, 670, 2286, 7918, 27770, 98424, 351983, 1268541, 4602752, 16799894, 61642078, 227239086, 841230292, 3126039364, 11656497518, 43601626146, 163561902392, 615183356156, 2319423532024, 8764535189296, 33187922345210, 125912855167740
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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G.f.: A(x) satisfies A^2*(x^2-2*x+2) - A*(x+1) + x = 0.
The generating function can be written as x/(1-x) times that of A082989.
G.f.: (2*x)/(1+x+(1-x)*sqrt(1-4*x)) = 1/(1-x(1-x)/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - Paul Barry, Jul 04 2009
a(n), n>0; is the upper left term in M^(n-1), where M is the infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
1, 1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, 1, ...
... (End)
D-finite with recurrence: 2*n*a(n) +4*(-3*n+4)*a(n-1) +(19*n-44)*a(n-2) + (-13*n + 36)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 24 2012
G.f: A = x/(1-(1-x)*x*C) = x*C/(1+x^2*C^2) = x*C^3/(1+2*x*C^3), where C is the g.f. of A000108.
A/x composed with x*C = g.f. of A165543, where A and C are as above. (End)
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EXAMPLE
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The a(5) = 6 cases are UUUUUDDDDD, UDUUUDUDDD, UDUUUDDUDD, UDUUDUUDDDD, UDUUDUDUDUDD and UUDDUUDUDD.
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MAPLE
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A:= proc(n) option remember; if n=0 then 0 else convert (series ((A(n-1)^2 *(x^2-2*x+2) +x)/ (x+1), x, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n): seq (a(n), n=0..25); # Alois P. Heinz, Aug 23 2008
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MATHEMATICA
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CoefficientList[Series[2*x/(1+x+(1-x)*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
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PROG
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(PARI) x='x+O('x^30); concat([0], Vec(2*x/(1+x+(1-x)*sqrt(1-4*x)))) \\ G. C. Greubel, Jan 15 2018
(Magma) /* Expansion */ Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 30); R!(2*x/(1+x+(1-x)*Sqrt(1-4*x))); // G. C. Greubel, Jan 15 2018
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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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STATUS
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approved
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