A126694 Expansion of g.f.: 1/(1 - 7*x*c(x)), where c(x) is the g.f. for A000108.

1, 7, 56, 455, 3710, 30282, 247254, 2019087, 16488710, 134656130, 1099686056, 8980749862, 73342721956, 598965319960, 4891549246290, 39947649057855, 326239122661830, 2664286127154330, 21758336553841440, 177693081299126610

Offset: 0

Author

Philippe Deléham, Feb 14 2007

Keywords

• nonn
• easy

Comments

The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.

Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from c(x) = 1/(1 - x*c(x)) == 1/(1 - 7*x*c(x)) (mod 2). - Peter Bala, Jul 24 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A000984, A007854, A076035, A076036, A127628, A115970.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // G. C. Greubel, May 05 2019

Mathematica

CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, May 05 2019 *)

PARI

my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019

Sage

(2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019

Formula

a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.

a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.

a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.

a(n) ~ 5 * 7^(2*n) / 6^(n+1). - Vaclav Kotesovec, Nov 29 2021

Extensions

a(16) corrected by G. C. Greubel, May 05 2019