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

0,2

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

Formula

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

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

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 *)

Prog

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

Crossrefs

Cf. A000108, A039599, A106566.

Sequence in context: A055272 A155196 A147838 * A111602 A299916 A091164

Adjacent sequences: A127625 A127626 A127627 * A127629 A127630 A127631

Keyword

nonn

Author

Paul Barry, Jan 20 2007

Status

approved