A370207 Number T(n,k) of unordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 6, 4, 3, 1, 1, 8, 10, 5, 3, 1, 1, 24, 18, 13, 6, 3, 1, 1, 30, 42, 23, 14, 6, 3, 1, 1, 74, 72, 55, 26, 15, 6, 3, 1, 1, 110, 146, 95, 61, 27, 15, 6, 3, 1, 1, 219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1, 309, 475, 351, 227, 115, 65, 28, 15, 6, 3, 1, 1

Offset

0,7

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

T(n,k) = (A370005(n,k) + A072233(n,k))/2.

Example

T(4,0) = 6: (1111,22), (1111,4), (211,4), (22,31), (22,4), (31,4).
T(4,1) = 4: (1111,31), (211,22), (211,31), (4,4).
T(4,2) = 3: (1111,211), (22,22), (31,31).
T(4,3) = 1: (211,211).
T(4,4) = 1: (1111,1111).
Triangle T(n,k) begins:
    1;
    0,   1;
    1,   1,   1;
    2,   2,   1,   1;
    6,   4,   3,   1,  1;
    8,  10,   5,   3,  1,  1;
   24,  18,  13,   6,  3,  1,  1;
   30,  42,  23,  14,  6,  3,  1, 1;
   74,  72,  55,  26, 15,  6,  3, 1, 1;
  110, 146,  95,  61, 27, 15,  6, 3, 1, 1;
  219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1;
  ...

Maple

b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
      add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*
       x^min(j, h)), h=0..m/i), j=0..n/i)))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(g(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= (n, k)-> (coeff(b(n$3), x, k)+coeff(g(n$2), x, k))/2:
seq(seq(T(n, k), k=0..n), n=0..12);

Crossrefs

Column k=0 gives A260669.

Row sums and T(2n,n) give A086737.

Cf. A000041, A000217, A008284, A072233, A370005.

Sequence in context: A014291 A136587 A136247 * A086610 A141760 A114626

Adjacent sequences: A370204 A370205 A370206 * A370208 A370209 A370210

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 11 2024

Status

approved