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A212698 Main transitions in systems of n particles with spin 3/2. 11
3, 24, 144, 768, 3840, 18432, 86016, 393216, 1769472, 7864320, 34603008, 150994944, 654311424, 2818572288, 12079595520, 51539607552, 219043332096, 927712935936, 3917010173952, 16492674416640, 69269232549888, 290271069732864, 1213860837064704 (list; graph; refs; listen; history; text; internal format)
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.

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(4x). - 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

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Last modified May 26 03:22 EDT 2017. Contains 287073 sequences.