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

%I A027471 #130 Jul 28 2025 21:40:18
%S A027471 0,1,6,27,108,405,1458,5103,17496,59049,196830,649539,2125764,6908733,
%T A027471 22320522,71744535,229582512,731794257,2324522934,7360989291,
%U A027471 23245229340,73222472421,230127770466,721764371007,2259436291848
%N A027471 a(n) = (n-1)*3^(n-2), n > 0.
%C A027471 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]
%C A027471 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
%C A027471 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
%C A027471 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
%C A027471 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
%H A027471 Vincenzo Librandi, <a href="/A027471/b027471.txt">Table of n, a(n) for n = 1..700</a>
%H A027471 Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1803.06706">Descent distribution on Catalan words avoiding a pattern of length at most three</a>, arXiv:1803.06706 [math.CO], 2018.
%H A027471 Samuele Giraudo, <a href="http://arxiv.org/abs/1603.01040">Pluriassociative algebras I: The pluriassociative operad</a>, arXiv:1603.01040 [math.CO], 2016.
%H A027471 Milan Janjić, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A027471 Milan Janjić and Boris Petković, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H A027471 Milan Janjić and Boris Petković, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H A027471 Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>
%H A027471 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=715">Encyclopedia of Combinatorial Structures 715</a>
%H A027471 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A027471 Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
%H A027471 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H A027471 Mark Shattuck, <a href="https://arxiv.org/abs/2502.10661">Enumeration of consecutive patterns in flattened Catalan words</a>, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.
%H A027471 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F A027471 From _Wolfdieter Lang_: (Start)
%F A027471 G.f.: (x/(1-3*x))^2.
%F A027471 E.g.f.: (1 + (3*x-1)*exp(3*x))/9.
%F A027471 a(n) = 3^(n-2)*(n-1) (convolution of A000244, powers of 3, with itself). (End)
%F A027471 a(n) = 6*a(n-1) - 9*a(n-2), n > 2, a(1)=0, a(2)=1. - _Barry E. Williams_, Jan 13 2000
%F A027471 a(n) = A036290(n-1)/3, for n>0. - _Paul Barry_, Feb 06 2004 [corrected by _Jerzy R Borysowicz_, Apr 03 2025]
%F A027471 a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
%F A027471 From _Paul Barry_, Feb 15 2005: (Start)
%F A027471 a(n) = (1/3)*Sum_{k=0..2n} T(n, k)*k, where T(n, k) is given by A027907.
%F A027471 a(n) = (1/3)*Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k).
%F A027471 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j-k).
%F A027471 a(n+1) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k+1). (End)
%F A027471 Sum_{n>=2} 1/a(n) = 3*log(3/2). - _Jaume Oliver Lafont_, Sep 19 2009
%F A027471 a(n) = 3*a(n-1) + 3^(n-2) (with a(1)=0). - _Vincenzo Librandi_, Dec 30 2010
%F A027471 Sum_{n>=2} (-1)^n/a(n) = 3*log(4/3). - _Amiram Eldar_, Oct 28 2020
%p A027471 seq((n-1)*3^(n-2), n=1..40); # _Muniru A Asiru_, Jul 15 2018
%t A027471 Table[(n-1)3^(n-2),{n,30}] (* or *)
%t A027471 LinearRecurrence[{6,-9},{0,1},30] (* _Harvey P. Dale_, Apr 14 2016 *)
%t A027471 Range[0, 24]! CoefficientList[ Series[x*Exp[3 x], {x, 0, 24}], x] (* _Robert G. Wilson v_, Aug 03 2018 *)
%o A027471 (PARI) a(n)=if(n<1, 0, (n-1)*3^(n-2));
%o A027471 (Magma) [(n-1)*3^(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 09 2011
%o A027471 (GAP) List([1..40], n-> (n-1)*3^(n-2)); # _Muniru A Asiru_, Jul 15 2018
%o A027471 (Sage) [3^(n-2)*(n-1) for n in (1..30)] # _G. C. Greubel_, May 20 2021
%Y A027471 Second column of A027465.
%Y A027471 Partial sums of A081038.
%Y A027471 Cf. A006234.
%K A027471 nonn,easy
%O A027471 1,3
%A A027471 _Olivier Gérard_
%E A027471 Edited by _Michael Somos_, Jul 10 2003