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A082301 G.f.: (1 - 4*x - sqrt(16*x^2 - 12*x + 1))/(2*x). 6
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.

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)

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

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<x>:=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 January 17 18:48 EST 2019. Contains 319251 sequences. (Running on oeis4.)