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 A121525 Number of up steps starting at an odd 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. 2
 0, 1, 5, 19, 67, 219, 690, 2110, 6322, 18639, 54268, 156398, 446960, 1268351, 3577679, 10039583, 28046201, 78039545, 216388938, 598136340, 1648730940, 4533180211, 12435470410, 34042090044, 93012717072, 253692955789 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n)=Sum(k*A121524(n,k), k=0..n-1). a(n)+A121523(n)=n*fibonacci(2n-1). LINKS Table of n, a(n) for n=1..26. E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217. Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1) FORMULA G.f.: z^2*(1-z-2z^2+3z^3-2z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)]. a(n) ~ (5-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+2)). - Vaclav Kotesovec, Mar 20 2014 EXAMPLE a(3)=5 because we have UDUDUD, UDU(U)DD, U(U)DDUD, U(U)D(U)DD and U(U)UDDD, the up steps starting at an odd level being shown between parentheses (U=(1,1), D=(1,-1)). MAPLE G:=z^2*(1-z-2*z^2+3*z^3-2*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): Gser:=series(G, z=0, 34): seq(coeff(Gser, z, n), n=1..30); MATHEMATICA Rest[CoefficientList[Series[x^2*(1-x-2*x^2+3*x^3-2*x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *) CROSSREFS Cf. A121523, A121524, A001519. Sequence in context: A067325 A273599 A347311 * A163872 A372884 A035344 Adjacent sequences: A121522 A121523 A121524 * A121526 A121527 A121528 KEYWORD nonn AUTHOR Emeric Deutsch, Aug 05 2006 STATUS approved

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Last modified July 14 15:30 EDT 2024. Contains 374322 sequences. (Running on oeis4.)