A201376 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs.

1, 1, 2, 2, 4, 9, 3, 7, 16, 31, 5, 12, 29, 57, 109, 7, 19, 47, 97, 189, 339, 11, 30, 77, 162, 323, 589, 1043, 15, 45, 118, 257, 522, 975, 1752, 2998, 22, 67, 181, 401, 831, 1576, 2876, 4987, 8406, 30, 97, 267, 608, 1279, 2472, 4571, 8043, 13715, 22652, 42, 139, 392, 907, 1941, 3804, 7128, 12693, 21893, 36535, 59521

Offset

0,3

Comments

By analogy with ordinary partitions (A000041). The empty partition gives T(0,0)=1 by definition. A201377 and A054225 give partitions of pairs into sums of 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..140, flattened

Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

Formula

For references, programs and g.f. see A054225.

Example

Partitions of (3,1) into positive pairs, T(3,1) = 7:
(3,1),
(3,0) + (0,1),
(2,1) + (1,0),
(2,0) + (1,1),
(2,0) + (1,0) + (0,1),
(1,1) + (1,0) + (1,0),
(1,0) + (1,0) + (1,0) + (0,1).
First ten rows of triangle:
0:                      1
1:                    1  2
2:                  2  4  9
3:                3  7  16  31
4:              5  12  29  57  109
5:            7  19  47  97  189  339
6:          11  30  77  162  323  589  1043
7:        15  45  118  257  522  975  1752  2998
8:      22  67  181  401  831  1576  2876  4987  8406
9:    30  97  267  608  1279  2472  4571  8043  13715  22652
X:  42  139  392  907  1941  3804  7128  12693  21893  36535  59521

Mathematica

max = 10; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max }, {k, 0, n}], {x, 0, 2max }, {y, 0, 2max }]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max }, {y, 0, 2max }], {x, y}]; t[n_, k_] := coes[[n+1, k+1]]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]] (* Jean-François Alcover, Dec 05 2011 *)
p = 2; q = 3; b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n] , 1|n]}]]; t[n_, k_] := b[p^n*q^k, p^n*q^k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

Prog

(Haskell) -- see link.

Crossrefs

T(n,0) = A000041(n);

T(1,k) = A000070(k), k <= 1; T(n,1) = A000070(n), n > 1;

T(2,k) = A000291(k), k <= 2; T(n,2) = A000291(n), n > 2;

T(3,k) = A000412(k), k <= 3; T(n,3) = A000412(n), n > 3;

T(4,k) = A000465(k), k <= 4; T(n,4) = A000465(n), n > 4;

T(5,k) = A000491(k), k <= 5; T(n,5) = A000491(n), n > 5;

T(6,k) = A002755(k), k <= 6; T(n,6) = A002755(n), n > 6;

T(7,k) = A002756(k), k <= 7; T(n,7) = A002756(n), n > 7;

T(8,k) = A002757(k), k <= 8; T(n,8) = A002757(n), n > 8;

T(9,k) = A002758(k), k <= 9; T(n,9) = A002758(n), n > 9;

T(10,k) = A002759(n), k <= 10; T(n,10) = A002759(n), n > 10;

T(n,n) = A002774(n).

See A054225 for another version.

Cf. A000041, A054242, A201377.

Sequence in context: A000301 A124439 A082836 * A005141 A220369 A220313

Adjacent sequences: A201373 A201374 A201375 * A201377 A201378 A201379

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Nov 30 2011

Extensions

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

Status

approved