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

1,1

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).

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).

a(n) = Sum_{i>=0} binomial(n,i)*i*3^i. - Geoffrey Critzer, Aug 08 2013

E.g.f.: 3*x*exp(4*x). - Geoffrey Critzer, Aug 08 2013

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

Mathematica

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

Prog

(PARI) mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212698.txt", n, " ", mtrans(n, 4)))
(Magma) [3*n*4^(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2015

Crossrefs

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

Cf. A002697, A002699, A133224.

Sequence in context: A056350 A056344 A201231 * A226511 A125651 A043017

Adjacent sequences: A212695 A212696 A212697 * A212699 A212700 A212701

Keyword

nonn,easy

Author

Stanislav Sykora, May 25 2012

Status

approved