|
|
A229724
|
|
Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).
|
|
1
|
|
|
1, 1, 2, 1, 2, 1, 4, 2, 1, 4, 3, 1, 7, 5, 2, 1, 7, 6, 3, 1, 12, 10, 5, 2, 1, 12, 12, 7, 3, 1, 19, 18, 11, 5, 2, 1, 19, 22, 14, 7, 3, 1, 30, 31, 21, 11, 5, 2, 1, 30, 37, 27, 15, 7, 3, 1, 45, 52, 38, 22, 11, 5, 2, 1, 45, 61, 48, 29, 15, 7, 3, 1, 67, 82, 66, 41
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
O.g.f. for column k: x^(2k-1)/[ prod_{j=1..2k-1}(1-x^j)*prod_{j>=k} (1-x^(2j)) ].
For even n=2j and k>=ceiling((n+2)/4) T(n,k)=A058695(j-k).
For odd n=2j-1 and k>=ceiling((n+2)/4) T(n,k)= A058696(j-k).
|
|
EXAMPLE
|
1;
1;
2, 1;
2, 1;
4, 2, 1;
4, 3, 1;
7, 5, 2, 1;
7, 6, 3, 1;
12, 10, 5, 2, 1;
12, 12, 7, 3, 1;
19, 18, 11, 5, 2, 1;
19, 22, 14, 7, 3, 1;
30, 31, 21, 11, 5, 2, 1;
T(7,2) = 5 because we have: 4+3 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1.
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, 1+x,
b(n, i-1) +`if`(i>n, 0, (p->`if`(irem(i, 2, 'r')=0, p,
coeff(p, x, 0)*(1+x^(r+1)) +add(coeff(p, x, j)*x^j,
j=r+2..degree(p))))(b(n-i, i)))))
end:
T:= n->(p-> seq(coeff(p, x, j), j=1..degree(p)))(b(n, n)):
|
|
MATHEMATICA
|
nn=16; Map[Select[#, #>0&]&, Drop[Transpose[Table[CoefficientList[Series[x^(2k-1)/Product[1-x^j, {j, 1, 2k-1}] /Product[(1-x^(2j)), {j, k, nn}], {x, 0, nn}], x], {k, 1, nn/2}]], 1]]//Grid
|
|
CROSSREFS
|
Column k=1 gives: A025065(n-1) for n>1.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|