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 A121485 Number of nondecreasing Dyck paths of semilength n and having no peaks at even level (n>=0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. 2
 1, 1, 2, 4, 8, 16, 33, 66, 136, 274, 562, 1138, 2327, 4725, 9645, 19613, 39997, 81397, 165906, 337773, 688260, 1401565, 2855432, 5815477, 11846941, 24129498, 49152840, 100116607, 203936639, 415394872, 846143795, 1723513075, 3510704795 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Column 0 of A121484. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 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 (1,4,-2,-4,0,1). FORMULA G.f.: z(1-z^2)(1-2z^2)/(1-z-4z^2+2z^3+4z^4-z^6). a(n) = a(n-1)+4*a(n-2)-2*a(n-3)-4*a(n-4)+a(n-6) for n>6. - Colin Barker, Sep 11 2015 EXAMPLE a(4)=4 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD and UUUDUDDD, where U=(1,1) and D=(1,-1). MAPLE G:=z*(1-z^2)*(1-2*z^2)/(1-4*z^2-z+4*z^4-z^6+2*z^3): Gser:=series(G, z=0, 40): seq(coeff(Gser, z, n), n=1..37); MATHEMATICA LinearRecurrence[{1, 4, -2, -4, 0, 1}, {1, 1, 2, 4, 8, 16}, 40] (* Vincenzo Librandi, Sep 12 2015 *) PROG (PARI) Vec(z*(1-z^2)*(1-2*z^2)/(1-z-4*z^2+2*z^3+4*z^4-z^6) + O(z^60)) \\ Michel Marcus, Sep 11 2015 (Magma) I:=[1, 1, 2, 4, 8, 16]; [n le 6 select I[n] else Self(n-1)+4*Self(n-2)-2*Self(n-3)-4*Self(n-4)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Sep 12 2015 CROSSREFS Cf. A121482, A121484. Sequence in context: A355548 A036373 A119610 * A308808 A324406 A182442 Adjacent sequences: A121482 A121483 A121484 * A121486 A121487 A121488 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Aug 02 2006 STATUS approved

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