A219557 -id:A219557 - cp's OEIS frontend

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.

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

Reinhard Zumkeller, Nov 30 2011

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.

			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

Alois P. Heinz, Rows n = 0..80, flattened
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

T(n,0) = A000009(n);
T(1,0) = A036469(0);  T(n,1) = A036469(n) for n > 0.
See A054242 for another version.
Cf. A000009, A054225, A201376.
T(n,n) = A219554(n). Row sums give: A219557. - Alois P. Heinz, Nov 22 2012

Haskell
```
-- see link.
```

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 *)

For g.f. see A054242.

Entry revised by N. J. A. Sloane, Nov 30 2011

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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.

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