A082301 G.f.: (1 - 4*x - sqrt(16*x^2 - 12*x + 1))/(2*x).

1, 5, 30, 205, 1530, 12130, 100380, 857405, 7503330, 66931030, 606337380, 5563370130, 51594699780, 482860844580, 4554484964280, 43252833007005, 413224841606130, 3968768817574030, 38297678538914580, 371128975862945030

Offset: 0

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

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.

Crossrefs

Cf. A006318, A047891.

Programs

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

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

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 *)

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

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