cp's OEIS Frontend

A212698 Main transitions in systems of n particles with spin 3/2.

%I A212698 #59 May 24 2025 10:34:17
%S A212698 3,24,144,768,3840,18432,86016,393216,1769472,7864320,34603008,
%T A212698 150994944,654311424,2818572288,12079595520,51539607552,219043332096,
%U A212698 927712935936,3917010173952,16492674416640,69269232549888,290271069732864,1213860837064704,5066549580791808
%N A212698 Main transitions in systems of n particles with spin 3/2.
%C A212698 Please refer to the general explanation in A212697. This particular sequence is obtained for base b=4, corresponding to spin S = (b-1)/2 = 3/2.
%C A212698 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the union of x and y for every (x,y) in B. [See Relation (28): U(n) in document of Ross La Haye in reference.] - _Bernard Schott_, Jan 04 2013
%C A212698 A002697 is the analogous sequence if "union" is replaced by "intersection" and A002699 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y and Y union X are considered as two distinct Cartesian products, if we want to consider that X Union Y = Y Union X are the same Cartesian product, see A133224. - _Bernard Schott_ Jan 11 2013
%H A212698 Stanislav Sykora, <a href="/A212698/b212698.txt">Table of n, a(n) for n = 1..100</a>
%H A212698 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 A212698 Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
%H A212698 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).
%F A212698 a(n) = n*(b-1)*b^(n-1). For this sequence, set b=4.
%F A212698 a(n) = 3*n*4^(n-1).
%F A212698 a(n) = 3*A002697(n).
%F A212698 From _Geoffrey Critzer_, Aug 08 2013: (Start)
%F A212698 a(n) = Sum_{i>=0} binomial(n,i)*i*3^i.
%F A212698 E.g.f.: 3*x*exp(4*x). (End)
%F A212698 G.f.: 3*x/(4*x-1)^2. - _Colin Barker_, Nov 03 2014
%F A212698 From _Elmo R. Oliveira_, May 24 2025: (Start)
%F A212698 a(n) = 8*a(n-1) - 16*a(n-2) for n > 2.
%F A212698 a(n) = A008585(n)*A000302(n-1). (End)
%t A212698 Table[Sum[Binomial[n,i] i 3^i,{i,0,n}],{n,1,21}] (* _Geoffrey Critzer_, Aug 08 2013 *)
%o A212698 (PARI) mtrans(n, b) = n*(b-1)*b^(n-1);
%o A212698 for (n=1, 100, write("b212698.txt", n, " ", mtrans(n, 4)))
%o A212698 (Magma) [3*n*4^(n-1): n in [1..30]]; // _Vincenzo Librandi_, Nov 29 2015
%Y A212698 Cf. A001787, A212697, A212699, A212700, A212701, A212702, A212703, A212704 (for b = 2, 3, 5, 6, 7, 8, 9, 10).
%Y A212698 Cf. A000302, A002697, A002699, A008585, A133224.
%K A212698 nonn,easy
%O A212698 1,1
%A A212698 _Stanislav Sykora_, May 25 2012