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A078009 a(0)=1, for n>=1 a(n)=sum(k=0,n,5^k*N(n,k)) where N(n,k) =1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). 11
1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076 (list; graph; refs; listen; history; text; internal format)
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(1-5x)/(1-4x); a(n+1) is the series reversion of x/(1+6x+5x^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 Schroder 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.

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

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

...

- Gary W. Adamson, Jul 08 2011

a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012

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

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, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)

PROG

(PARI) a(n)=sum(k=0, n, 5^k/n*binomial(n, k)*binomial(n, k+1))

CROSSREFS

Cf. A001003, A007564, A059231, A127848.

Sequence in context: A152107 A143023 * A127848 A113573 A083161 A077147

Adjacent sequences:  A078006 A078007 A078008 * A078010 A078011 A078012

KEYWORD

nonn

AUTHOR

Benoit Cloitre, May 10 2003

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.