

A285852


Rectangular array read by rows: T(n,k) is the number of words of length n on alphabet {0,1,2} that have exactly k records, n>=0, 0<=k<=3.


1



1, 0, 0, 0, 0, 3, 0, 0, 0, 6, 3, 0, 0, 14, 12, 1, 0, 36, 39, 6, 0, 98, 120, 25, 0, 276, 363, 90, 0, 794, 1092, 301, 0, 2316, 3279, 966, 0, 6818, 9840, 3025, 0, 20196, 29523, 9330, 0, 60074, 88572, 28501, 0, 179196, 265719, 86526
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,6


COMMENTS

A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i<j.


LINKS

Table of n, a(n) for n=0..51.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009.


FORMULA

G.f.: Product_{j=1..3} (1 + y*x/(1  j*x)). Generally for words on alphabet {0,1,...,r} the o.g.f. is Product_{j=1..r} (1 + y*x/(1  j*x)).


EXAMPLE

1, 0, 0, 0;
0, 3, 0, 0;
0, 6, 3, 0;
0, 14, 12, 1;
0, 36, 39, 6;
0, 98, 120, 25;
0, 276, 363, 90;
0, 794, 1092, 301;
0, 2316, 3279, 966;


MATHEMATICA

nn = 12; CoefficientList[Series[Product[1 + u z/(1  j z), {j, 1, 3}], {z, 0, nn}], {z, u}] // Grid


CROSSREFS

Column k=0 gives A000007.
Column k=1 gives A001550.
Column k=2 gives A029858.
Column k=3 gives A000392.
Row sums give A000244.
Sequence in context: A221702 A084681 A261038 * A096528 A161837 A299163
Adjacent sequences: A285849 A285850 A285851 * A285853 A285854 A285855


KEYWORD

nonn,tabf


AUTHOR

Geoffrey Critzer, Apr 27 2017


STATUS

approved



