A082302 Expansion of g.f.: (1 - 5*x - sqrt(25*x^2 - 14*x + 1))/(2*x).

1, 6, 42, 330, 2814, 25422, 239442, 2326434, 23151030, 234784662, 2417832186, 25216231866, 265796560302, 2827138163550, 30306009654690, 327081253546770, 3551148743559270, 38758882760119590, 425024567305557450

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*x) are given by a(0)=1 and, for n > 0, a(n) = (1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1).

Hankel transform is 6^C(n+1,2). - Philippe Deléham, Feb 11 2009

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.
• Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

Crossrefs

Cf. A006318, A047891.

Programs

GAP

Concatenation([1],List([1..20],n->(1/n)*Sum([0..n],k->6^k*Binomial(n,k)*Binomial(n,k-1)))); # Muniru A Asiru, Apr 05 2018

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-5*x-Sqrt(25*x^2-14*x+1))/(2*x))); // G. C. Greubel, Aug 16 2018

Maple

A082302_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 6*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A082302_list(18); # Peter Luschny, May 19 2011
a := n -> `if`(n=0, 1, 6*hypergeom([1 - n, -n], [2], 6)):
seq(simplify(a(n)), n=0..18); # Peter Luschny, May 22 2017

Mathematica

Table[SeriesCoefficient[(1-5*x-Sqrt[25*x^2-14*x+1])/(2*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PARI

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

PARI

x='x+O('x^99); Vec((1-5*x-(25*x^2-14*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018

Formula

Equals 6*A078018(n) for n > 0.

a(0)=1; for n > 0, a(n) = (1/n)*Sum_{k=0..n} 6^k*C(n, k)*C(n, k-1).

D-finite with recurrence: (n+1)*a(n) + 7*(1-2n)*a(n-1) + 25*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 08 2011

a(n) ~ sqrt(12 + 7*sqrt(6))*(7 + 2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

a(n) = 6*hypergeom([1 - n, -n], [2], 6) for n > 0. - Peter Luschny, May 22 2017

G.f.: 1/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018