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 A082301 G.f.: (1 - 4*x - sqrt(16*x^2 - 12*x + 1))/(2*x). 7
 1, 5, 30, 205, 1530, 12130, 100380, 857405, 7503330, 66931030, 606337380, 5563370130, 51594699780, 482860844580, 4554484964280, 43252833007005, 413224841606130, 3968768817574030, 38297678538914580, 371128975862945030 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally, coefficients of (1 - m*x - sqrt(m^2*x^2 - (2*m + 4)*x + 1))/(2*x) are given by a(0)=1 and, for n > 0, a(n) = (1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1). Hankel transform is 5^C(n+1,2). - Philippe Deléham, Feb 11 2009 Series reversion of x(1-x)/(1+4x). - Paul Barry, Oct 22 2009 a(n) is the number of Schroder paths of semilength n in which the (2,0)-steps come in 4 colors. Example: a(2)=30 because, denoting U=(1,1), H=(2,0), D=(1,-1), we have 4^2=16 paths of shape HH, 4 paths of shape HUD, 4 paths of shape UDH, 4 paths of shape UHD, and 1 path of each of the shapes UDUD, UUDD. - Emeric Deutsch, May 02 2011 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8. FORMULA a(0)=1; for n > 0, a(n) = (1/n)*Sum_{k=0..n} 5^k*C(n, k)*C(n, k-1). From Paul Barry, Oct 22 2009: (Start) D-finite with recurrence: a(n) = if(n=0, 1, if(n=1, 5, 6*((2n-1)/(n+1))*a(n-1)-16*((n-2)/(n+1))*a(n-2))). a(n) = A078009(n)*(5 - 4*0^n). (End) a(n) ~ sqrt(10 + 6*sqrt(5))*(6 + 2*sqrt(5))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012. Equivalently, a(n) ~ 5^(1/4) * 2^(2*n) * phi^(2*n + 1) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021 a(n) = 5*hypergeom([1 - n, -n], [2], 5) for n > 0. - Peter Luschny, May 22 2017 G.f.: 1/(1 - 4*x - x/(1 - 4*x - x/(1 - 4*x - x/(1 - 4*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018 MAPLE A082301_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; for w from 1 to n do a[w] := 5*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A082301_list(19); # Peter Luschny, May 19 2011 a := n -> `if`(n=0, 1, 5*hypergeom([1 - n, -n], [2], 5)): seq(simplify(a(n)), n=0..19); # Peter Luschny, May 22 2017 MATHEMATICA Table[SeriesCoefficient[(1-4*x-Sqrt[16*x^2-12*x+1])/(2*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *) PROG (PARI) a(n)=if(n<1, 1, sum(k=0, n, 5^k*binomial(n, k)*binomial(n, k-1))/n) (PARI) x='x+O('x^99); Vec((1-4*x-(16*x^2-12*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018 (MAGMA) Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1 -4*x-Sqrt(16*x^2-12*x+1))/(2*x))) // G. C. Greubel, Feb 10 2018 (GAP) Concatenation([1], List([1..20], n->(1/n)*Sum([0..n], k->5^k*Binomial(n, k)*Binomial(n, k-1)))); # Muniru A Asiru, Apr 05 2018 CROSSREFS Cf. A006318, A047891. Sequence in context: A091927 A253076 A165312 * A144180 A222050 A091122 Adjacent sequences:  A082298 A082299 A082300 * A082302 A082303 A082304 KEYWORD nonn AUTHOR Benoit Cloitre, May 10 2003 STATUS approved

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Last modified August 9 00:15 EDT 2022. Contains 356016 sequences. (Running on oeis4.)