A081038 3rd binomial transform of (1,2,0,0,0,0,0,0,...).

1, 5, 21, 81, 297, 1053, 3645, 12393, 41553, 137781, 452709, 1476225, 4782969, 15411789, 49424013, 157837977, 502211745, 1592728677, 5036466357, 15884240049, 49977243081, 156905298045, 491636600541, 1537671920841

Offset: 0

Author

Paul Barry, Mar 03 2003

Keywords

• easy
• nonn

Comments

a(n) is the number of distinguished parts in all compositions of n+1 in which some (possibly all or none) of the parts have been distinguished. a(1) = 2 because we have: 2', 1'+1, 1+1', 1'+1' where we see 5's marking the distinguished parts. With offset=1, a(n) = Sum_{k=1..n} A200139(n,k)*k. - Geoffrey Critzer, Jan 12 2013

For n>=1, a(n-1) the number of ternary strings of length 2n containing the block 11..12 with n ones where no runs of length larger than n are permitted. - Marko Riedel, Mar 08 2016

Binomial transform of {A001787(n + 1)}{n >= 0}. - _Wolfdieter Lang, Oct 01 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.
• Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Cf. A001787, A001792, A081039, A081040, A086972, A200139.

First differences of A027471.

Cf. A269914, A269915, A269916, A269917.

Programs

Magma

[(2*n+3)*3^(n-1): n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

Maple

A081038:=n->(2*n+3)*3^(n-1): seq(A081038(n), n=0..30); # Wesley Ivan Hurt, Mar 07 2016

Mathematica

LinearRecurrence[{6,-9},{1,5},40] (* Harvey P. Dale, Jun 22 2012 *)

Formula

G.f.: (1-x)/(1-3*x)^2.

a(n) = 6*a(n-1) - 9*a(n-2), with a(0)=1, a(1)=5.

a(n) = (2*n+3)*3^(n-1).

a(n) = Sum_{k=0..n} (k+1)*2^k*binomial(n, k).

a(n) = 2*A086972(n) - 1. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 10 2008

From Amiram Eldar, May 17 2022: (Start)

Sum_{n>=0} 1/a(n) = 9*(sqrt(3)*arctanh(1/sqrt(3)) - 1).

Sum_{n>=0} (-1)^n/a(n) = 9 - 3*sqrt(3)*Pi/2. (End)

E.g.f.: exp(3*x)*(1 + 2*x). - Stefano Spezia, Jan 31 2025