OFFSET
0,3
COMMENTS
By analogy with ordinary partitions into distinct parts (A000009). The empty partition gives T(0,0)=1 by definition. A201376 and A054242 give partitions of pairs into sums of not necessarily distinct pairs.
Parts (i,j) are "positive" in the sense that min {i,j} >= 0 and max {i,j} >0. The empty partition of (0,0) is counted as 1.
LINKS
Alois P. Heinz, Rows n = 0..80, flattened
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377
FORMULA
For g.f. see A054242.
EXAMPLE
Partitions of (2,1) into distinct positive pairs, T(2,1) = 3:
(2,1),
(2,0) + (0,1),
(1,1) + (1,0);
Partitions of (2,2) into distinct positive pairs, T(2,2) = 5:
(2,2),
(2,1) + (0,1),
(2,0) + (0,2),
(1,2) + (1,0),
(1,1) + (1,0) + (0,1).
First ten rows of triangle:
0: 1
1: 1 2
2: 1 3 5
3: 2 5 9 17
4: 2 7 14 27 46
5: 3 10 21 42 74 123
6: 4 14 31 64 116 197 323
7: 5 19 44 93 174 303 506 809
8: 6 25 61 132 254 452 769 1251 1966
9: 8 33 83 185 363 659 1141 1885 3006 4660
MATHEMATICA
nmax = 10;
f[x_, y_] := Product[1 + x^n y^k, {n, 0, nmax}, {k, 0, nmax}]/2;
se = Series[f[x, y], {x, 0, nmax}, {y, 0, nmax}];
coes = CoefficientList[se, {x, y}];
t[n_ /; n >= 0, k_] /; 0 <= k <= n := coes[[n-k+1, k+1]];
T[n_, k_] := t[n+k, k];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2021 *)
PROG
(Haskell) -- see link.
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Nov 30 2011
EXTENSIONS
Entry revised by N. J. A. Sloane, Nov 30 2011
STATUS
approved