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

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

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.

Crossrefs

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

Programs

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

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 *)

PARI

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

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

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

Extensions

Corrected by T. D. Noe, Oct 25 2006