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A201376 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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..55, 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

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Last modified April 10 11:35 EDT 2021. Contains 342845 sequences. (Running on oeis4.)