A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15

Offset

0,7

Comments

Compare with A284592.

Links

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

H. S. Wilf, Lectures on Integer Partitions

Formula

O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A032302.

Example

Square array begins
  n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009
  1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765
  2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744
  3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18
  4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19
  5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28
  6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33
  7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41
  ...
T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).

Maple

# A284593 as a square array
ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Aug 24 2019

Mathematica

nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
T[n_, k_] := M[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Crossrefs

Cf. A000009 (row 0), A096765 (row 1), A015744 (row 2), A032302 (antidiagonal sums).

Cf. A284592, A322210.

Main diagonal gives 2*A108796 (for n>0).

Sequence in context: A115236 A307777 A365661 * A190672 A327910 A242998

Adjacent sequences: A284590 A284591 A284592 * A284594 A284595 A284596

Keyword

nonn,tabl,easy

Author

Peter Bala, Mar 30 2017

Status

approved