A082148 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^k*N(n,k), where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

1, 1, 11, 131, 1661, 22101, 305151, 4335711, 63009881, 932449961, 14004694451, 212944033051, 3271618296661, 50711564152381, 792088104593511, 12454801769554551, 196991734871121201, 3131967533789345361, 50026642742943415131, 802406215117502069811

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 10^C(n+1,2). - Philippe Deléham, Oct 29 2007

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

1, 1;

10, 10, 10;

1, 1, 1, 1;

10, 10, 10, 10, 10;

1, 1, 1, 1, 1, 1;

...

- Gary W. Adamson, Jul 08 2011

Shifts left when INVERT transform applied ten times. - Benedict W. J. Irwin, Feb 07 2016

For fixed m > 0, if g.f. = (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) then a(n,m) ~ (m + 2 + 2*sqrt(m+1))^(n + 1/2) / (2*sqrt(Pi) * (m+1)^(3/4) * n^(3/2)). - Vaclav Kotesovec, Mar 19 2018

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, A007564, A059231.

Programs

Magma

I:=[1,11]; [1] cat [n le 2 select I[n] else (11*(2*n-1)*Self(n-1) - 81*(n-2)*Self(n-2))/(n+1): n in [1..30]]; // G. C. Greubel, Feb 10 2018

Maple

A082148_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]+10*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A082148_list(17); # Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+9*x-Sqrt[81*x^2-22*x+1])/(20*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Sum[10^k*1/n*Binomial[n, k]*Binomial[n, k + 1], {k, 0, n}];
a[0] = 1; Array[a, 20, 0] (* Robert G. Wilson v, Feb 10 2018 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 10];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,10^k/n*binomial(n,k)*binomial(n,k+1)))

Formula

G.f.: (1+9*x-sqrt(81*x^2-22*x+1))/(20*x).

a(n) = Sum_{k=0..n} A088617(n, k)*10^k*(-9)^(n-k). - Philippe Deléham, Jan 21 2004

a(n) = (11*(2n-1)*a(n-1) - 81*(n-2)*a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005

a(n) ~ sqrt(20+11*sqrt(10))*(11+2*sqrt(10))^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - x/(1 - 10*x/(1 - x/(1 - 10*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017

a(n) = hypergeom([1 - n, -n], [2], 10). - Peter Luschny, Mar 19 2018