A081178 a(0) = 1; for n>=1, a(n) = Sum_{k=0..n} 7^k*N(n,k), where N(n,k)=(1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248, 99362894, 1143498224, 13326176998, 156950554384, 1865210341828, 22338852956064, 269355965364459, 3267146912972328, 39837475762660374, 488032452193307568

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 7^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

7, 7, 7

1, 1, 1, 1

7, 7, 7, 7, 7

1, 1, 1, 1, 1, 1

...

(End)

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

G.f.: 1/(1 - x/(1 - 7*x/(1 - x/(1 - 7*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

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.

Programs

Magma

B:=Binomial;
A081178:= func< n | n eq 0 select 1 else (&+[7^k*B(n,k)*B(n,k+1): k in [0..n]])/n >;
[A081178(n): n in [0..40]]; // G. C. Greubel, Jan 15 2024

Maple

A081178_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]+7*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A081178_list(18); # Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+6*x-Sqrt[36*x^2-16*x+1])/(14*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 7];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,7^k/n*binomial(n,k)*binomial(n,k+1)))

SageMath

def A081178(n):
    b=binomial;
    if n==0: return 1
    else: return (1/n)*sum(7^k*b(n,k)*b(n,k+1) for k in range(n+1))
[A081178(n) for n in range(41)] # G. C. Greubel, Jan 15 2024

Formula

G.f.: (1+6*x-sqrt(36*x^2-16*x+1))/(14*x).

a(n) = (8*(2*n-1)*a(n-1) - 36*(n-2)*a(n-2))/(n+1) for n>=2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005

a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n/(14*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012

a(n) = hypergeom([1 - n, -n], [2], 7). - Peter Luschny, Mar 19 2018