A078009 a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263).

1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076

Offset

0,3

Comments

More generally coefficients of (1 + m*x - sqrt(m^2*x^2 - (2*m+2)*x + 1) )/( 2*m*x ) are given by a(n) = Sum_{k=0..n} (m+1)^k * N(n,k).

a(n) is the series reversion of x*(1-5*x)/(1-4*x). a(n+1) is the series reversion of x/(1 + 6*x + 5*x^2). a(n+1) counts (6,5)-Motzkin paths of length n, where there are 6 colors available for the H(1,0) steps and 5 for the U(1,1) steps. - Paul Barry, May 19 2005

The Hankel transform of this sequence is 5^C(n+1,2). - Philippe Deléham, Oct 29 2007

a(n) is the number of Schröder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 4 colors. Example: a(2)=6 because we have UDUD, UUDD, UBD, UGD, URD, and UYD, where U=(1,1), D=(1,-1), while B, G, R, and Y are, respectively, blue, green, red, and yellow (2,0)-steps. - Emeric Deutsch, May 02 2011

Shifts left when INVERT transform applied five times. - Benedict W. J. Irwin, Feb 03 2016

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.

Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.

Formula

G.f.: (1 + 4*x - sqrt(16*x^2 - 12*x + 1))/(10*x).

a(n) = Sum_{k=0..n} A088617(n, k)*5^k*(-4)^(n-k). - Philippe Deléham, Jan 21 2004

With offset 1 : a(1)=1, a(n) = -4*a(n-1) + 5*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004

a(n+1) = Sum_{k=0..floor(n/2)} C(n, 2*k)*C(k)*6^(n-2k)*5^k; - Paul Barry, May 19 2005

a(n) = ( 6*(2*n-1)*a(n-1) - 16*(n-2)*a(n-2) ) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005

From Gary W. Adamson, Jul 08 2011: (Start)

a(n) = upper left term in M^n, M = the production matrix:

1, 1

5, 5, 5

1, 1, 1, 1

5, 5, 5, 5, 5

1, 1, 1, 1, 1, 1

... (End)

a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012. Equivalently, a(n) ~ 2^(2*n) * phi^(2*n + 1) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

a(n) = A127848(n) for n > 0. - Philippe Deléham, Apr 03 2013

G.f.: 1/(1 - x/(1 - 5*x/(1 - x/(1 - 5*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

a(n) = hypergeom([1 - n, -n], [2], 5). - Peter Luschny, Mar 19 2018

Maple

A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *)

Prog

(PARI) a(n)=sum(k=0, n, 5^k/n*binomial(n, k)*binomial(n, k+1))
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019
(Magma) [1] cat [&+[5^k*Binomial(n, k)*Binomial(n, k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020
(Sage) a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019

Crossrefs

Cf. A001003, A007564, A059231, A127848.

Sequence in context: A186654 A152107 A143023 * A127848 A113573 A083161

Adjacent sequences: A078006 A078007 A078008 * A078010 A078011 A078012

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved