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 A112412 Number of Dyck paths of semilength n for which the number of ascents of length 1 is equal to the number of descents of length 1. 0
 1, 1, 2, 5, 12, 30, 82, 237, 708, 2188, 6980, 22814, 75994, 257266, 883006, 3065757, 10748620, 38005844, 135385700, 485439532, 1750738084, 6347006468, 23118315044, 84565309214, 310536661002, 1144393816154, 4231119156334 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Apparently: Number of Dyck n-paths with equal numbers of peaks to the left and to the right of the midpoint (ordinate x=n). - David Scambler, Aug 08 2012 REFERENCES R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 179. LINKS FORMULA G.f. is the diagonal of g(t, s, z), where g=g(t, s, z) is defined by z(1+tz-tsz)(1+sz-tsz)g^2 - [1+(1-ts)z-(1-t)(1-s)z^2]g+1=0 (g is the trivariate g.f. of Dyck paths, where z marks semilength and t (s) marks number of ascents (descents) of length 1. EXAMPLE a(4)=12 because among the 14 Dyck paths of semilength 4 the only counterexamples are UUDUUDDD and UUUDDUDD, where U=(1,1), D=(1,-1). PROG (PARI) z=x; s; t; f(g) = z*(1+t*z-t*s*z)*(1+s*z-t*s*z)*g^2-(1+(1-t*s)*z-(1-t)*(1-s)*z^2)*g+1 nxt(fx) = fx=truncate(fx); fx+=O(x^2)*x^poldegree(fx); fx+=f(fx) oo=30; g=1+O(z); for(n=1, oo, g=nxt(g)); g1=polcoeff(subst(subst(g, s, y), t, 1/y), 0, y); for(n=0, oo, print(n" "polcoeff(g1, n))) CROSSREFS Sequence in context: A179544 A182488 A261788 * A309506 A319557 A261937 Adjacent sequences:  A112409 A112410 A112411 * A112413 A112414 A112415 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 08 2005 STATUS approved

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Last modified May 9 13:42 EDT 2021. Contains 343742 sequences. (Running on oeis4.)