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A201376
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Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs.
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5
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Rows n = 0..55, flattened
Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377
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FORMULA
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For references, programs and g.f. see A054225.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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Haskell: see link.
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Reinhard Zumkeller, Nov 30 2011
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Nov 30 2011
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STATUS
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approved
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