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A127628 G.f. 1/(1-6*x*c(x)) where c(x) is the g.f. of A000108. 7
1, 6, 42, 300, 2154, 15492, 111492, 802584, 5778090, 41600532, 299517996, 2156509416, 15526797252, 111792690600, 804906480840, 5795323452720, 41726317225770, 300429441596340, 2163091823919900, 15574260559056840 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified June 24 06:17 EDT 2021. Contains 345416 sequences. (Running on oeis4.)