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

0,3

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 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.

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

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 *)

Prog

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

Crossrefs

Cf. A001003, A007564, A059231.

Sequence in context: A076357 A015606 A077417 * A075509 A061113 A261689

Adjacent sequences: A082145 A082146 A082147 * A082149 A082150 A082151

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved