A127628 G.f. 1/(1-6*x*c(x)) where c(x) is the g.f. of A000108.

1, 6, 42, 300, 2154, 15492, 111492, 802584, 5778090, 41600532, 299517996, 2156509416, 15526797252, 111792690600, 804906480840, 5795323452720, 41726317225770, 300429441596340, 2163091823919900, 15574260559056840, 112134673904493420, 807369644235408120

Offset: 0

Author

Paul Barry, Jan 20 2007

Keywords

• nonn

Comments

Image of 6^n under the Catalan transform g(x)->g(xc(x)). The Hankel transform of this sequence and of the aerated version with g.f. 1/(1-6*x^2*c(x^2)) is 6^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n.

Links

• Robert Israel, Table of n, a(n) for n = 0..1165

Crossrefs

Cf. A000108, A039599, A106566.

Programs

Maple

f:= gfun:-rectoproc({(72 + 144*n)*a(n) + (-82 - 56*n)*a(n + 1) + (5*n + 10)*a(n + 2), a(0) = 1, a(1) = 6},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 28 2020

Mathematica

RecurrenceTable[{(-56*n - 82)*a[n + 1] + (5*n +  10)*a[n + 2] + (144*n + 72)*a[n] == 0, a[0] == 1, a[1] == 6}, a, {n, 0, 50}] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

PARI

my(x='x+O('x^25)); Vec(1/(1-6*x*(1-sqrt(1-4*x))/(2*x))) \\ Michel Marcus, Sep 15 2022

Formula

a(n) = 1 if n=0, Sum_{k=1..n} C(2*n-k-1,n-k)*k*6^k/n otherwise;

a(n) = Sum_{k=0..n} C(2*n,n-k)*(2*k+1)*5^k/(n+k+1).

a(n) = Sum_{k=0..n} A106566(n,k)*6^k. - Philippe Deléham, Feb 04 2007

a(n) = Sum_{k=0..n} A039599(n,k)*5^k. - Philippe Deléham, Sep 08 2007

a(n) = (36*a(n-1) - 6*A000108(n-1))/5 for n >= 1, a(0) = 1. - Philippe Deléham, Nov 27 2007

Conjecture: 5*n*a(n) + 2*(15-28*n)*a(n-1) + 72*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

G.f.: (2+3*sqrt(1-4*x))/(5-36*x). Mathar's conjecture verified using the differential equation (144*x^2-56*x+5)*y'+(72*x-26)*y = 4 satisfied by the g.f. - Robert Israel, Aug 28 2020