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A082181 a(0)=1; for n>=1, a(n) = sum(k=0..n, 9^k*N(n,k)), where N(n,k) =1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). 7
1, 1, 10, 109, 1270, 15562, 198100, 2596645, 34825150, 475697854, 6595646860, 92590323058, 1313427716380, 18798095833012, 271118225915560, 3936516861402901, 57494017447915150, 844109420603623030 (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+4)*x+1))/((2*m+2)*x) are given by: a(n) = sum(k=0..n, (m+1)^k*N(n,k)).

The Hankel transform of this sequence is 9^C(n+1,2). - Philippe Deléham, Oct 29 2007

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

1, 1

9, 9, 9

1, 1, 1, 1

9, 9, 9, 9, 9

1, 1, 1, 1, 1, 1

...

- Gary W. Adamson, Jul 08 2011

Shifts left when INVERT transform applied nine times. - Benedict W. J. Irwin, Feb 07 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+8*x-sqrt(64*x^2-20*x+1))/(18*x).

a(n) = Sum_{k=0..n} A088617(n, k)*9^k*(-8)^(n-k). - Philippe Deléham, Jan 21 2004

a(n) = (10(2n-1)a(n-1) - 64(n-2)a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005

a(n) ~ 2^(4*n+1)/(3*sqrt(3*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - x/(1 - 9*x/(1 - x/(1 - 9*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

MAPLE

A082181_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]+9*add(a[j]*a[w-j-1], j=1..w-1) od;

convert(a, list) end: A082181_list(17); # Peter Luschny, May 19 2011

MATHEMATICA

Table[SeriesCoefficient[(1+8*x-Sqrt[64*x^2-20*x+1])/(18*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PROG

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 9^k/n*binomial(n, k)*binomial(n, k+1)))

CROSSREFS

Cf. A001003, A007564, A059231.

Sequence in context: A015591 A078922 A199760 * A190919 A095740 A075508

Adjacent sequences:  A082178 A082179 A082180 * A082182 A082183 A082184

KEYWORD

nonn

AUTHOR

Benoit Cloitre, May 10 2003

EXTENSIONS

Corrected by T. D. Noe, Oct 25 2006

STATUS

approved

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Last modified September 24 04:27 EDT 2017. Contains 292403 sequences.