|
Search: id:A116664
|
|
|
| A116664 |
|
Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k parts that appear exactly once (n>=0, k>=0). |
|
+0
2
|
|
| 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 4, 0, 2, 2, 2, 3, 4, 0, 1, 3, 4, 3, 0, 3, 7, 1, 1, 5, 4, 6, 0, 4, 10, 2, 2, 6, 7, 9, 0, 7, 12, 5, 3, 7, 13, 11, 0, 1, 8, 18, 7, 5, 0, 11, 15, 18, 1, 1, 10, 25, 11, 8, 0, 13, 23, 24, 2, 2, 15, 32, 16, 13, 0, 16, 33, 32, 5, 3, 18, 43, 24, 19, 0, 23, 40
(list; graph; listen)
|
|
|
OFFSET
|
0,12
|
|
|
COMMENT
|
Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0)=A097242(n). Sum(k*T(n,k), k>=0)=A116665(n).
|
|
FORMULA
|
G.f.=product(1+tx^(2j-1)+x^(4j-2)/(1-x^(2j-1)), j=1..infinity).
|
|
EXAMPLE
|
T(10,2)=3 because the only partitions of 10 into odd parts and having exactly 2 parts that appear only once are [9,1],[7,3], and [5,3,1,1].
Triangle starts:
1;
0,1;
1,0;
1,1;
1,0,1;
1,2,0;
2,1,1;
1,4,0;
|
|
MAPLE
|
g:=product(1+t*x^(2*j-1)+x^(2*(2*j-1))/(1-x^(2*j-1)), j=1..30): gser:=simplify(series(g, x=0, 30)): P[0]:=1: for n from 1 to 25 do P[n]:=coeff(gser, x^n) od: for n from 0 to 25 do seq(coeff(P[n], t, j), j=0..floor(sqrt(n))) od; # yields sequence in triangular form, with one extra 0 in some rows
|
|
CROSSREFS
|
Cf. A000009, A097242, A116665.
Adjacent sequences: A116661 A116662 A116663 this_sequence A116665 A116666 A116667
Sequence in context: A111330 A117447 A053250 this_sequence A024161 A035156 A063883
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2006
|
|
|
Search completed in 0.002 seconds
|
