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A121486
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Number of peaks at even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
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2
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0, 1, 4, 13, 43, 132, 400, 1184, 3461, 9999, 28634, 81383, 229860, 645731, 1805582, 5028189, 13952221, 38590922, 106434540, 292792026, 803565215, 2200694791, 6015268164, 16412564173, 44708036568, 121600924117, 330277253560
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..27.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1).
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FORMULA
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a(n) = Sum(k*A121484(n,k),k=0..n-1).
G.f.: z^2*(1-z)(1-z-3z^2+3z^3-z^4)/[(1+z)(1-z-z^2)(1-3z+z^2)^2].
a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5 * 2^(n+2)). - Vaclav Kotesovec, Mar 20 2014
20*a(n) = -8*(-1)^n +10*(2*A001871(n)-5*A001871(n-1))+5*(4*A000045(n+1)-7*A000045(n))-3*(4*A001906(n+1)+9*A001906(n)). - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(3)=4 because in UDUDUD, UDUU|DD, UU|DDUD, UU|DU|DD and UUUDDD we have altogether 4 peaks at even level (shown by a |); here U=(1,1) and D=(1,-1).
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MAPLE
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G:=z^2*(1-z)*(1-z-3*z^2+3*z^3-z^4)/(1+z)/(1-z-z^2)/(1-3*z+z^2)^2: Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=1..30);
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MATHEMATICA
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Rest[CoefficientList[Series[x^2*(1-x)*(1-x-3*x^2+3*x^3-x^4)/(1+x)/(1-x-x^2)/(1-3*x+x^2)^2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)
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CROSSREFS
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Cf. A121484, A121483, A038731.
Sequence in context: A072307 A355116 A266494 * A339063 A188176 A003688
Adjacent sequences: A121483 A121484 A121485 * A121487 A121488 A121489
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Aug 02 2006
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STATUS
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approved
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