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

3, 24, 144, 768, 3840, 18432, 86016, 393216, 1769472, 7864320, 34603008, 150994944, 654311424, 2818572288, 12079595520, 51539607552, 219043332096, 927712935936, 3917010173952, 16492674416640, 69269232549888, 290271069732864, 1213860837064704, 5066549580791808

Offset: 1

Author

Stanislav Sykora, May 25 2012

Keywords

• nonn
• easy

Comments

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.

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

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

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..100
• 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.
• Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
• Index entries for linear recurrences with constant coefficients, signature (8,-16).

Crossrefs

Cf. A001787, A212697, A212699, A212700, A212701, A212702, A212703, A212704 (for b = 2, 3, 5, 6, 7, 8, 9, 10).

Cf. A000302, A002697, A002699, A008585, A133224.

Programs

Magma

[3*n*4^(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2015

Mathematica

Table[Sum[Binomial[n,i] i 3^i,{i,0,n}],{n,1,21}] (* Geoffrey Critzer, Aug 08 2013 *)

PARI

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212698.txt", n, " ", mtrans(n, 4)))

Formula

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=4.

a(n) = 3*n*4^(n-1).

a(n) = 3*A002697(n).

From Geoffrey Critzer, Aug 08 2013: (Start)

a(n) = Sum_{i>=0} binomial(n,i)*i*3^i.

E.g.f.: 3*x*exp(4*x). (End)

G.f.: 3*x/(4*x-1)^2. - Colin Barker, Nov 03 2014

From Elmo R. Oliveira, May 24 2025: (Start)

a(n) = 8*a(n-1) - 16*a(n-2) for n > 2.

a(n) = A008585(n)*A000302(n-1). (End)