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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.

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