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A201377 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of distinct pairs.

%I A201377 #47 May 11 2023 18:34:11
%S A201377 1,1,2,1,3,5,2,5,9,17,2,7,14,27,46,3,10,21,42,74,123,4,14,31,64,116,
%T A201377 197,323,5,19,44,93,174,303,506,809,6,25,61,132,254,452,769,1251,1966,
%U A201377 8,33,83,185,363,659,1141,1885,3006,4660
%N A201377 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of partitions of (n,k) into a sum of distinct pairs.
%C A201377 By analogy with ordinary partitions into distinct parts (A000009). The empty partition gives T(0,0)=1 by definition. A201376 and A054242 give partitions of pairs into sums of not necessarily distinct pairs.
%C A201377 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 A201377 Alois P. Heinz, <a href="/A201377/b201377.txt">Rows n = 0..80, flattened</a>
%H A201377 Reinhard Zumkeller, <a href="/A054225/a054225_1.lhs.txt">Haskell programs for A054225, A054242, A201376, A201377</a>
%F A201377 For g.f. see A054242.
%e A201377 Partitions of (2,1) into distinct positive pairs, T(2,1) = 3:
%e A201377 (2,1),
%e A201377 (2,0) + (0,1),
%e A201377 (1,1) + (1,0);
%e A201377 Partitions of (2,2) into distinct positive pairs, T(2,2) = 5:
%e A201377 (2,2),
%e A201377 (2,1) + (0,1),
%e A201377 (2,0) + (0,2),
%e A201377 (1,2) + (1,0),
%e A201377 (1,1) + (1,0) + (0,1).
%e A201377 First ten rows of triangle:
%e A201377 0:                      1
%e A201377 1:                    1  2
%e A201377 2:                  1  3  5
%e A201377 3:                2  5  9  17
%e A201377 4:              2  7  14  27  46
%e A201377 5:            3  10  21  42  74  123
%e A201377 6:          4  14  31  64  116  197  323
%e A201377 7:        5  19  44  93  174  303  506  809
%e A201377 8:      6  25  61  132  254  452  769  1251  1966
%e A201377 9:    8  33  83  185  363  659  1141  1885  3006  4660
%t A201377 nmax = 10;
%t A201377 f[x_, y_] := Product[1 + x^n y^k, {n, 0, nmax}, {k, 0, nmax}]/2;
%t A201377 se = Series[f[x, y], {x, 0, nmax}, {y, 0, nmax}];
%t A201377 coes = CoefficientList[se, {x, y}];
%t A201377 t[n_ /; n >= 0, k_] /; 0 <= k <= n := coes[[n-k+1, k+1]];
%t A201377 T[n_, k_] := t[n+k, k];
%t A201377 Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 08 2021 *)
%o A201377 (Haskell) -- see link.
%Y A201377 T(n,0) = A000009(n);
%Y A201377 T(1,0) = A036469(0);  T(n,1) = A036469(n) for n > 0.
%Y A201377 See A054242 for another version.
%Y A201377 Cf. A000009, A054225, A201376.
%Y A201377 T(n,n) = A219554(n). Row sums give: A219557. - _Alois P. Heinz_, Nov 22 2012
%K A201377 nonn,tabl
%O A201377 0,3
%A A201377 _Reinhard Zumkeller_, Nov 30 2011
%E A201377 Entry revised by _N. J. A. Sloane_, Nov 30 2011