A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401, 226739489, 2878666857, 37006326777, 480750990993, 6301611631473, 83240669582937, 1106980509493641, 14808497812637121, 199138509770855489, 2690461489090104009

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

Shifts left when INVERT transform applied eight 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 + 7*x - sqrt(49*x^2-18*x+1))/(16*x).

a(n) = Sum_{k=0..n} A088617(n, k)*8^k*(-7)^(n-k). - Philippe Deléham, Jan 21 2004

a(n) = (9(2n-1)a(n-1) - 49(n-2)a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005

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

1, 1

8, 8, 8

1, 1, 1, 1

8, 8, 8, 8, 8

1, 1, 1, 1, 1, 1

...

- Gary W. Adamson, Jul 08 2011

a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - x/(1 - 8*x/(1 - x/(1 - 8*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

a(n) = hypergeom([1 - n, -n], [2], 8). - Peter Luschny, Mar 19 2018

Maple

A082147_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]+8*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A082147_list(18); # Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+7*x-Sqrt[49*x^2-18*x+1])/(16*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
f[n_] := Sum[ 8^k*Binomial[n, k]*Binomial[n, k + 1]/n, {k, 0, n}]; f[0] = 1; Array[f, 21, 0] (* Robert G. Wilson v, Feb 24 2018 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 8];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

Prog

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 8^k/n*binomial(n, k)*binomial(n, k+1)))
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1+7*x-Sqrt(49*x^2-18*x+1))/(16*x))) // G. C. Greubel, Feb 05 2018
(GAP) a:=n->Sum([0..n], k->8^k*(1/n)*Binomial(n, k)*Binomial(n, k+1));;
Concatenation([1], List([1..18], n->a(n))); # Muniru A Asiru, Feb 10 2018

Crossrefs

Cf. A001003, A007564, A059231.

Sequence in context: A109002 A142991 A152267 * A095722 A199759 A069573

Adjacent sequences: A082144 A082145 A082146 * A082148 A082149 A082150

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved