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

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 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.

Crossrefs

Cf. A001003, A007564, A059231.

Programs

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

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1+7*x-Sqrt(49*x^2-18*x+1))/(16*x))) // G. C. Greubel, Feb 05 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 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,8^k/n*binomial(n,k)*binomial(n,k+1)))

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