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, 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

Offset: 1

Author

Geoffrey Critzer, Sep 28 2013

Keywords

• nonn
• tabf

Comments

Row sums are A086543.

Examples

			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.
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Column k=1 gives: A025065(n-1) for n>1.

Programs

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)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 28 2013

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

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).