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A081178 a(0)=1; for n>=1, a(n) = sum(7^k*N(n,k), k=0..n), where N(n,k)=1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). 8
1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248, 99362894, 1143498224, 13326176998, 156950554384, 1865210341828, 22338852956064, 269355965364459, 3267146912972328, 39837475762660374, 488032452193307568 (list; graph; refs; listen; history; text; internal format)
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

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

...

- Gary W. Adamson, Jul 08 2011

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(2n-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

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 *)

PROG

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

CROSSREFS

Cf. A001003, 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

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Last modified November 18 19:06 EST 2017. Contains 294894 sequences.