A027471 a(n) = (n-1)*3^(n-2), n > 0.

0, 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, 649539, 2125764, 6908733, 22320522, 71744535, 229582512, 731794257, 2324522934, 7360989291, 23245229340, 73222472421, 230127770466, 721764371007, 2259436291848

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Arithmetic derivative of 3^(n-1): a(n) = A003415(A000244(n-1)). - Reinhard Zumkeller, Feb 26 2002 [Offset corrected by Jianing Song, May 28 2024]

Binomial transform of A053220(n+1) is a(n+2). Binomial transform of A001787 is a(n+1). Binomial transform of A045883(n-1). - Michael Somos, Jul 10 2003

If X_1,X_2,...,X_n are 3-blocks of a (3n+1)-set X then, for n >= 1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). > - Milan Janjic, Nov 18 2007

Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n+1) = the sum of the differences in size (i.e., |y|-|x|) for all (x, y) of S. - Ross La Haye, Nov 19 2007

Number of substrings 00 (or 11, or 22) in all ternary words of length n: a(3) = 6 because we have 000, 001, 002, 100, 200 (with 000 contributing two substrings). - Darrell Minor, Jul 17 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..700
• Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
• Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
• Milan Janjić, Two Enumerative Functions
• Milan Janjić and Boris Petković, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Frank Ellermann, Illustration of binomial transforms
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 715
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Second column of A027465.

Partial sums of A081038.

Cf. A006234.

Programs

GAP

List([1..40], n-> (n-1)*3^(n-2)); # Muniru A Asiru, Jul 15 2018

Magma

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

Maple

seq((n-1)*3^(n-2), n=1..40); # Muniru A Asiru, Jul 15 2018

Mathematica

Table[(n-1)3^(n-2),{n,30}] (* or *)
LinearRecurrence[{6,-9},{0,1},30] (* Harvey P. Dale, Apr 14 2016 *)
Range[0, 24]! CoefficientList[ Series[x*Exp[3 x], {x, 0, 24}], x] (* Robert G. Wilson v, Aug 03 2018 *)

PARI

a(n)=if(n<1, 0, (n-1)*3^(n-2));

Sage

[3^(n-2)*(n-1) for n in (1..30)] # G. C. Greubel, May 20 2021

Formula

From Wolfdieter Lang: (Start)

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

E.g.f.: (1 + (3*x-1)*exp(3*x))/9.

a(n) = 3^(n-2)*(n-1) (convolution of A000244, powers of 3, with itself). (End)

a(n) = 6*a(n-1) - 9*a(n-2), n > 2, a(1)=0, a(2)=1. - Barry E. Williams, Jan 13 2000

a(n) = A036290(n-1)/3, for n>0. - Paul Barry, Feb 06 2004 [corrected by Jerzy R Borysowicz, Apr 03 2025]

a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.

From Paul Barry, Feb 15 2005: (Start)

a(n) = (1/3)*Sum_{k=0..2n} T(n, k)*k, where T(n, k) is given by A027907.

a(n) = (1/3)*Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k).

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j-k).

a(n+1) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k+1). (End)

Sum_{n>=2} 1/a(n) = 3*log(3/2). - Jaume Oliver Lafont, Sep 19 2009

a(n) = 3*a(n-1) + 3^(n-2) (with a(1)=0). - Vincenzo Librandi, Dec 30 2010

Sum_{n>=2} (-1)^n/a(n) = 3*log(4/3). - Amiram Eldar, Oct 28 2020

Extensions

Edited by Michael Somos, Jul 10 2003