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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..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

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