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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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Terrence Fine, Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359.
Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, 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
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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