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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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.

Cf. A000009, A054225, A201376.

T(n,n) = A219554(n). Row sums give: A219557. - Alois P. Heinz, Nov 22 2012

Sequence in context: A210867 A019588 A193953 * A322942 A060083 A069931

Adjacent sequences:  A201374 A201375 A201376 * A201378 A201379 A201380

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, Nov 30 2011

EXTENSIONS

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

STATUS

approved

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Last modified January 19 22:12 EST 2022. Contains 350466 sequences. (Running on oeis4.)