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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.

%I A001559 M3497 N1418 #59 May 19 2024 14:02:53
%S A001559 1,4,15,54,193,690,2476,8928,32358,117866,431381,1585842,5853849,
%T A001559 21690378,80650536,300845232,1125555054,4222603968,15881652606,
%U A001559 59873283372,226214536506,856431978324,3248562071800,12344168149224,46984664348488,179114048943078
%N 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.
%C A001559 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
%D A001559 Terrence Fine, Extrapolation when very little is known about the source. Information and Control 16 (1970), 331-359.
%D A001559 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
%D A001559 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001559 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001559 T. D. Noe, <a href="/A001559/b001559.txt">Table of n, a(n) for n = 0..200</a>
%H A001559 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001559 Simon Plouffe, <a href="http://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
%H A001559 D. G. Rogers, <a href="https://doi.org/10.1016/0097-3165(77)90082-6">Similarity relations on finite ordered sets</a>, J. Combin. Theory, Series A, Vol. 23, No. 1 (1977), pp. 88-98. Erratum, loc. cit., Vol. 25 (1978), pp. 95-96.
%F A001559 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
%F A001559 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
%F A001559 (n+4)*a(n) = (-(15/2)*n + 4)*a(n-3) + ((11/2)*n + 12)*a(n-1)
%F A001559 + (-4*n + 3)*a(n-2) + (-2*n + 3)*a(n-4). - _Simon Plouffe_, Feb 09 2012
%F A001559 a(n) ~ 7*2^(2*n+6)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%F A001559 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
%e A001559 G.f. = 1 + 4*x + 15*x^2 + 54*x^3 + 193*x^4 + 690*x^5 + 2476*x^6 + 8928*x^7 + ...
%t A001559 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 *)
%t A001559 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 *)
%o A001559 (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 */
%o A001559 (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
%o A001559 (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
%K A001559 nonn
%O A001559 0,2
%A A001559 _N. J. A. Sloane_
%E A001559 Better definition and more terms from _Michael Somos_, Jul 14 2009