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

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 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.

Crossrefs

Cf. A001003, A007564, A059231.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 + 5*x - Sqrt(25*x^2-14*x+1))/(12*x) )); // G. C. Greubel, Jun 29 2019

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 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,6^k/n*binomial(n,k)*binomial(n,k+1)))

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

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