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A201377
Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of distinct pairs.
11
1, 1, 2, 1, 3, 5, 2, 5, 9, 17, 2, 7, 14, 27, 46, 3, 10, 21, 42, 74, 123, 4, 14, 31, 64, 116, 197, 323, 5, 19, 44, 93, 174, 303, 506, 809, 6, 25, 61, 132, 254, 452, 769, 1251, 1966, 8, 33, 83, 185, 363, 659, 1141, 1885, 3006, 4660
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.
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.
CROSSREFS
T(n,0) = A000009(n);
T(1,0) = A036469(0); T(n,1) = A036469(n) for n > 0.
See A054242 for another version.
T(n,n) = A219554(n). Row sums give: A219557. - Alois P. Heinz, Nov 22 2012
Sequence in context: A210867 A019588 A193953 * A368070 A322942 A060083
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Nov 30 2011
EXTENSIONS
Entry revised by N. J. A. Sloane, Nov 30 2011
STATUS
approved