A078018 a(n) = Sum_{k=0..n} 6^k*N(n,k), with a(0)=1, where N(n,k) = C(n,k) * C(n,k+1)/n are the Narayana numbers (A001263).

1, 1, 7, 55, 469, 4237, 39907, 387739, 3858505, 39130777, 402972031, 4202705311, 44299426717, 471189693925, 5051001609115, 54513542257795, 591858123926545, 6459813793353265, 70837427884259575, 780073647992404615

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

Shifts left when INVERT transform applied six 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 + 5*x - sqrt(25*x^2-14*x+1))/(12*x).

a(n) = Sum_{k=0..n} A088617(n, k)*6^k*(-5)^(n-k). - Philippe Deléham, Jan 21 2004

a(n) = ( 7*(2*n-1)*a(n-1) - 25*(n-2)*a(n-2) ) / (n+1) for n>=2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005

From Gary W. Adamson, Jul 08 2011: (Start)

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

1, 1;

6, 6, 6;

1, 1, 1, 1;

6, 6, 6, 6, 6;

1, 1, 1, 1, 1, 1;

... (End)

a(n) ~ sqrt(12+7*sqrt(6))*(7+2*sqrt(6))^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012

G.f.: 1/(1 - x/(1 - 6*x/(1 - x/(1 - 6*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

a(n) = hypergeom([1 - n, -n], [2], 6). - Peter Luschny, Mar 19 2018

Maple

A078018_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]+6*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A078018_list(19);
# Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+5*x-Sqrt[25*x^2-14*x+1])/(12*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_]:= Hypergeometric2F1[1 - n, -n, 2, 6]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Mar 19 2018 *)

Prog

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 6^k/n*binomial(n, k)*binomial(n, k+1)))
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 + 5*x - Sqrt(25*x^2-14*x+1))/(12*x) )); // G. C. Greubel, Jun 29 2019
(Sage) a=((1 + 5*x - sqrt(25*x^2-14*x+1))/(12*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 29 2019

Crossrefs

Cf. A001003, A007564, A059231.

Sequence in context: A113714 A246459 A152262 * A357207 A108628 A116862

Adjacent sequences: A078015 A078016 A078017 * A078019 A078020 A078021

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved