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A027471 a(n) = (n-1)*3^(n-2), n > 0. 39
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Arithmetic derivative of 3^n: a(n) = A003415(A000244(n)). - Reinhard Zumkeller, Feb 26 2002

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..700

Jean-Luc Baril, Sergey Kirgizov, 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 Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.

F. 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.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

FORMULA

From Wolfdieter Lang: (Start)

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

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

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

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

a(n) = A036290(n)/3. - Paul Barry, Feb 06 2004

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)

Numerators of sequence a[ 2, n ] in (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

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

MAPLE

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

MATHEMATICA

Join[{a=0, b=1}, Table[c=6*b-9*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

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 *)

PROG

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

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

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

CROSSREFS

Second column of A027465.

Cf. A006234.

Partial sums of A081038.

Sequence in context: A099623 A119852 A220529 * A305780 A037695 A318638

Adjacent sequences:  A027468 A027469 A027470 * A027472 A027473 A027474

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Edited by Michael Somos, Jul 10 2003

STATUS

approved

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Last modified February 20 18:39 EST 2019. Contains 320345 sequences. (Running on oeis4.)