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%I #61 Sep 08 2022 08:44:52
%S 0,3,18,81,324,1215,4374,15309,52488,177147,590490,1948617,6377292,
%T 20726199,66961566,215233605,688747536,2195382771,6973568802,
%U 22082967873,69735688020,219667417263,690383311398,2165293113021,6778308875544
%N a(n) = n*3^n.
%C If X_1,X_2,...,X_n is a partition of a 3n-set X into 3-blocks then, for n>0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - _Milan Janjic_, Jul 21 2007
%H Vincenzo Librandi, <a href="/A036290/b036290.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F From _Paul Barry_, Feb 06 2004: (Start)
%F A trinomial transform. Differentiate (1+x+x^2)^n and set x=1.
%F a(n) = Sum_{i=0..n} Sum_{j=0..n} (2*n-2*i-j)*n!/(i!*j!*(n-i-j)!). (End)
%F From _Paul Barry_, Feb 15 2005: (Start)
%F a(n) = Sum_{k=0..2*n} T(n, k)*k, where T(n, k) is given by A027907.
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k). (End)
%F From _R. J. Mathar_, Jun 19 2011: (Start)
%F G.f.: 3*x / (3*x-1)^2 .
%F a(n) = 3*A027471(n+1). (End)
%F Sum_{n>=1} 1/a(n) = log(3/2) = 0.405465108... = A016578. - _Franz Vrabec_, Jan 07 2012
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(4/3) = A083679. - _Amiram Eldar_, Jul 20 2020
%F a(n) = 6*a(n-1)-9*a(n-2). - _Wesley Ivan Hurt_, Apr 26 2021
%p A036290 := proc(n) n*3^n ; end proc: # _R. J. Mathar_, Jun 18 2011
%t nn=20;a=1/(1-3x);CoefficientList[Series[x D[ a,x] ,{x,0,nn}],x] (* _Geoffrey Critzer_, Nov 18 2012 *)
%t Table[n 3^n, {n, 0, 30}] (* _Vincenzo Librandi_, Jul 06 2017 *)
%o (PARI) a(n)=3^n*n \\ _Charles R Greathouse IV_, Jun 18, 2011
%o (Magma) [n*3^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 06 2017
%Y Cf. A006234, A016578, A027471, A083679, A289399 (partial sums).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.
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