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).

1, 1, 10, 109, 1270, 15562, 198100, 2596645, 34825150, 475697854, 6595646860, 92590323058, 1313427716380, 18798095833012, 271118225915560, 3936516861402901, 57494017447915150, 844109420603623030

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

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

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

... (End)

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*(2*n-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

a(n) = hypergeom([1 - n, -n], [2], 9). - Peter Luschny, Mar 19 2018

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 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 9];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

Prog

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 9^k/n*binomial(n, k)*binomial(n, k+1)))
(Magma) [(&+[Binomial(n, k)*Binomial(n-1, k)*9^k/(k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2022
(SageMath) [sum(binomial(n, k)*binomial(n-1, k)*9^k/(k+1) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2022

Crossrefs

Cf. A001003, A001263, A007564, A059231, A088617.

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