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

0,3

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 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.

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

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 *)

Prog

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

Crossrefs

Cf. A001003, A001263, A007564, A059231.

Sequence in context: A187709 A292865 A152265 * A096341 A199687 A225033

Adjacent sequences: A081175 A081176 A081177 * A081179 A081180 A081181

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved