|
Search: id:A114088
|
|
|
| A114088 |
|
Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n>=1; 0<=k<=n-1). |
|
+0
4
|
|
| 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 5, 6, 6, 5, 3, 2, 1, 1, 1, 6, 8, 8, 7, 5, 3, 2, 1, 1, 1, 7, 10, 10, 9, 7, 5, 3, 2, 1, 1, 1, 9, 13, 13, 12, 10, 7, 5, 3, 2, 1, 1, 1, 10, 16, 17, 15, 13, 10, 7, 5, 3, 2, 1, 1, 1, 12, 20, 22, 20, 17
(list; table; graph; listen)
|
|
|
OFFSET
|
1,7
|
|
|
COMMENT
|
Row sums yield A000041. Column 0 is A003114. Sum(k*T(n,k),k=0..n-1)=A114089(n).
|
|
REFERENCES
|
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
|
|
FORMULA
|
G.f.=sum(q^(k^2)/product((1-q^j)(1-tq^j),j=1..k),k=1..infinity).
|
|
EXAMPLE
|
T(7,2)=3 because we have [5,1,1], [3,2,1,1], and [2,2,2,1] (the bottom tails are [1,1], [1,1], and [2,1], respectively).
Triangle starts:
1;
1,1;
1,1,1;
2,1,1,1;
2,2,1,1,1;
3,3,2,1,1,1;
3,4,3,2,1,1,1;
|
|
MAPLE
|
g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j), j=1..k), k=1..20): gserz:=simplify(series(g, z=0, 30)): for n from 1 to 14 do P[n]:=coeff(gserz, z^n) od: for n from 1 to 14 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A115994, A115995, A114087, A116365, A114089.
Sequence in context: A137350 A114087 A008284 this_sequence A037306 A007424 A085424
Adjacent sequences: A114085 A114086 A114087 this_sequence A114089 A114090 A114091
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006
|
|
|
Search completed in 0.002 seconds
|
