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%I #53 May 11 2023 18:32:22
%S 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,
%T 323,589,1043,15,45,118,257,522,975,1752,2998,22,67,181,401,831,1576,
%U 2876,4987,8406,30,97,267,608,1279,2472,4571,8043,13715,22652,42,139,392,907,1941,3804,7128,12693,21893,36535,59521
%N Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of pairs.
%C 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.
%C 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.
%H Alois P. Heinz, <a href="/A201376/b201376.txt">Rows n = 0..140, flattened</a>
%H Reinhard Zumkeller, <a href="/A054225/a054225_1.lhs.txt">Haskell programs for A054225, A054242, A201376, A201377</a>
%F For references, programs and g.f. see A054225.
%e Partitions of (3,1) into positive pairs, T(3,1) = 7:
%e (3,1),
%e (3,0) + (0,1),
%e (2,1) + (1,0),
%e (2,0) + (1,1),
%e (2,0) + (1,0) + (0,1),
%e (1,1) + (1,0) + (1,0),
%e (1,0) + (1,0) + (1,0) + (0,1).
%e First ten rows of triangle:
%e 0: 1
%e 1: 1 2
%e 2: 2 4 9
%e 3: 3 7 16 31
%e 4: 5 12 29 57 109
%e 5: 7 19 47 97 189 339
%e 6: 11 30 77 162 323 589 1043
%e 7: 15 45 118 257 522 975 1752 2998
%e 8: 22 67 181 401 831 1576 2876 4987 8406
%e 9: 30 97 267 608 1279 2472 4571 8043 13715 22652
%e X: 42 139 392 907 1941 3804 7128 12693 21893 36535 59521
%t 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 *)
%t 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_ *)
%o (Haskell) -- see link.
%Y T(n,0) = A000041(n);
%Y T(1,k) = A000070(k), k <= 1; T(n,1) = A000070(n), n > 1;
%Y T(2,k) = A000291(k), k <= 2; T(n,2) = A000291(n), n > 2;
%Y T(3,k) = A000412(k), k <= 3; T(n,3) = A000412(n), n > 3;
%Y T(4,k) = A000465(k), k <= 4; T(n,4) = A000465(n), n > 4;
%Y T(5,k) = A000491(k), k <= 5; T(n,5) = A000491(n), n > 5;
%Y T(6,k) = A002755(k), k <= 6; T(n,6) = A002755(n), n > 6;
%Y T(7,k) = A002756(k), k <= 7; T(n,7) = A002756(n), n > 7;
%Y T(8,k) = A002757(k), k <= 8; T(n,8) = A002757(n), n > 8;
%Y T(9,k) = A002758(k), k <= 9; T(n,9) = A002758(n), n > 9;
%Y T(10,k) = A002759(n), k <= 10; T(n,10) = A002759(n), n > 10;
%Y T(n,n) = A002774(n).
%Y See A054225 for another version.
%Y Cf. A000041, A054242, A201377.
%K nonn,tabl
%O 0,3
%A _Reinhard Zumkeller_, Nov 30 2011
%E Entry revised by _N. J. A. Sloane_, Nov 30 2011
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