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 A001559 a(0) = 1, a(1) = 4; thereafter a(n)*(2n + 10) - a(n-1)*(11n + 35) + a(n-2)*(8n + 2) + a(n-3)*(15n + 7) + a(n-4)*(4n - 2) = 0. (Formerly M3497 N1418) 4
 1, 4, 15, 54, 193, 690, 2476, 8928, 32358, 117866, 431381, 1585842, 5853849, 21690378, 80650536, 300845232, 1125555054, 4222603968, 15881652606, 59873283372, 226214536506, 856431978324, 3248562071800, 12344168149224, 46984664348488, 179114048943078 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Apparently, the number of hill-free Dyck (n+4)-paths with at least two returns. E.g., the a(1)=4 hill-free 5-paths are UUUDDDUUDD, UUDUDDUUDD, UUDDUUUDDD and UUDDUUDUDD with 2 returns each. - David Scambler, Aug 26 2012 REFERENCES Fine, Terrence; Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359. Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012 D. G. Rogers, Similarity relations on finite ordered sets, J. Combin. Theory, A 23 (1977), 88-98. Erratum, loc. cit., 25 (1978), 95-96. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..200 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics. FORMULA 0 = -a(n) * n * (2*n + 10) * (7*n + 13) + a(n-1) * (49*n^3 + 252*n^2 + 419*n + 240) + a(n-2) * (2*n + 2) * (2*n + 3) * (7*n + 20). - Michael Somos, Jul 14 2009 G.f.: 2 / (1 - 4*x + x^2 + 2*x^3 + (1 - 2*x - x^2) * sqrt(1 - 4*x )). - Michael Somos, Jul 14 2009 (n + 4) a(n) = (- 15/2 n + 4) a(n - 3) + (11/2 n + 12) a(n - 1) + (- 4 n + 3) a(n - 2) + (- 2 n + 3) a(n - 4). [Simon Plouffe, Feb 09 2012] a(n) ~ 7*2^(2*n+6)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013 0 = a(n) * (-112336*a(n+1) - 30270*a(n+2) - 88504*a(n+3) - 845858*a(n+4) + 217516*a(n+5)) + a(n+1) * (-14042*a(n+1) + 440283*a(n+2) - 328994*a(n+3) - 731173*a(n+4) + 230486*a(n+5)) + a(n+2) * (38900*a(n+2) - 812130*a(n+3) + 1877788*a(n+4) - 386672*a(n+5)) + a(n+3) * (-535412*a(n+3) - 86596*a(n+4) + 44840*a(n+5)) if n>-3. - Michael Somos, Apr 03 2014 EXAMPLE G.f. = 1 + 4*x + 15*x^2 + 54*x^3 + 193*x^4 + 690*x^5 + 2476*x^6 + 8928*x^7 + ... MATHEMATICA nn = 20; a[-2] = 0; a[-1] = 0; a[0] = 1; a[1] = 4; Do[a[n] = (a[n - 1]*(11*n + 35) - a[n - 2]*(8*n + 2) - a[n - 3]*(15*n + 7) - a[n - 4]*(4*n - 2))/(2*n + 10), {n, 2, nn}]; Table[a[n], {n, 0, nn}] (* T. D. Noe, May 09 2012 *) CoefficientList[Series[2/(1-4*x+x^2+2*x^3 +(1-2*x-x^2)*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, Apr 28 2019 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 - 4*x + x^2 + 2*x^3 + (1 - 2*x - x^2) * sqrt(1 - 4*x + x*O(x^n))), n))}; /* Michael Somos, Jul 14 2009 */ (MAGMA) R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(1-4*x +x^2+2*x^3 +(1-2*x-x^2)*Sqrt(1-4*x)) )); // G. C. Greubel, Apr 28 2019 (Sage) (2/(1-4*x+x^2+2*x^3 +(1-2*x-x^2)*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019 CROSSREFS Sequence in context: A006234 A094821 A071723 * A002311 A102349 A219603 Adjacent sequences:  A001556 A001557 A001558 * A001560 A001561 A001562 KEYWORD nonn AUTHOR EXTENSIONS Better definition and more terms from Michael Somos, Jul 14 2009 STATUS approved

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Last modified October 22 09:56 EDT 2019. Contains 328315 sequences. (Running on oeis4.)