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

0,2

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

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

Mathematica

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

Prog

(PARI) my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 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

Crossrefs

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

Sequence in context: A152776 A155197 A147839 * A305198 A264912 A323216

Adjacent sequences: A126691 A126692 A126693 * A126695 A126696 A126697

Keyword

nonn,easy

Author

Philippe Deléham, Feb 14 2007

Extensions

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

Status

approved