A146523 Binomial transform of A010685.

1, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Author

Philippe Deléham, Oct 30 2008

Keywords

• nonn
• easy

Comments

Linked to A029609 by a Catalan transform.

Hankel transform is (1, -15, 0, 0, 0, 0, 0, 0, 0, ...).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A010685, A020714, A029609, A084215, A109466.

Programs

Mathematica

CoefficientList[Series[(1+3x)/(1-2x), {x,0,50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 5*2^(Range[40] -1)] (* G. C. Greubel, Nov 23 2021 *)

PARI

a(n)=if(n,5<<(n-1),1) \\ Charles R Greathouse IV, Jan 17 2012

Sage

[1]+[5*2^(n-1) for n in (1..50)] # G. C. Greubel, Nov 23 2021

Formula

a(n) = 5*2^(n-1) for n >= 1, a(0) = 1.

a(n) = Sum_{k=0..n} A109466(n,k)*A029609(k).

a(n) = A084215(n+1) = A020714(n-1), n > 0. - R. J. Mathar, Nov 02 2008

G.f.: (1 + 3*x)/(1 - 2*x). - Vladimir Joseph Stephan Orlovsky, Jun 21 2011

G.f.: G(0), where G(k)= 1 + 3*x/(1 - 2*x/(2*x + 3*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013

E.g.f.: (5*exp(2*x) - 3)/2. - Stefano Spezia, Feb 20 2023