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A001559
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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)
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4
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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]
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
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EXAMPLE
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G.f. = 1 + 4*x + 15*x^2 + 54*x^3 + 193*x^4 + 690*x^5 + 2476*x^6 + 8928*x^7 + ...
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MATHEMATICA
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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 *)
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PROG
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(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<x>:=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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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