|
Search: id:A117147
|
|
|
| A117147 |
|
Triangle read by rows: T(n,k) is the number of partitions of n with k parts in which no part occurs more than 3 times (n>=1, k>=1). |
|
+0
2
|
|
| 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 4, 3, 1, 1, 4, 5, 4, 2, 1, 4, 7, 6, 3, 1, 1, 5, 8, 9, 5, 1, 1, 5, 10, 11, 8, 3, 1, 6, 12, 14, 11, 5, 1, 1, 6, 14, 18, 15, 8, 2, 1, 7, 16, 23, 20, 11, 4, 1, 7, 19, 27, 27, 17, 6, 1, 1, 8, 21, 33, 34, 23, 10, 2, 1, 8, 24, 39, 43, 32, 15, 4, 1, 9
(list; graph; listen)
|
|
|
OFFSET
|
1,8
|
|
|
COMMENT
|
Row n has floor(sqrt(6n+6)-3/2) terms. Row sums yield A001935. Sum(k*T(n,k),k>=0)=A117148(n).
|
|
FORMULA
|
G.f.=G(t,x)= -1+product(1+tx^j+t^2*x^(2j)+t^3*x^(3j), j=1..infinity).
|
|
EXAMPLE
|
T(7,3)=4 because we have [5,1,1],[4,2,1],[3,3,1], and [3,2,2].
Triangle starts:
1;
1,1;
1,1,1;
1,2,1;
1,2,2,1;
1,3,3,2;
1,3,4,3,1;
|
|
MAPLE
|
g:=-1+product(1+t*x^j+t^2*x^(2*j)+t^3*x^(3*j), j=1..35): gser:=simplify(series(g, x=0, 23)): for n from 1 to 18 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 18 do seq(coeff(P[n], t^j), j=1..floor(sqrt(6*n+6)-3/2)) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A001935, A117148.
Sequence in context: A122632 A134542 A106254 this_sequence A111007 A103691 A103441
Adjacent sequences: A117144 A117145 A117146 this_sequence A117148 A117149 A117150
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 07 2006
|
|
|
Search completed in 0.002 seconds
|
