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
KEYWORD
nonn,easy
AUTHOR
Stanislav Sykora, May 25 2012
STATUS
approved