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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 distinct pairs.
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%I #47 May 11 2023 18:34:11

%S 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 197,323,5,19,44,93,174,303,506,809,6,25,61,132,254,452,769,1251,1966,

%U 8,33,83,185,363,659,1141,1885,3006,4660

%N 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 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 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="/A201377/b201377.txt">Rows n = 0..80, flattened</a>

%H Reinhard Zumkeller, <a href="/A054225/a054225_1.lhs.txt">Haskell programs for A054225, A054242, A201376, A201377</a>

%F For g.f. see A054242.

%e Partitions of (2,1) into distinct positive pairs, T(2,1) = 3:

%e (2,1),

%e (2,0) + (0,1),

%e (1,1) + (1,0);

%e Partitions of (2,2) into distinct positive pairs, T(2,2) = 5:

%e (2,2),

%e (2,1) + (0,1),

%e (2,0) + (0,2),

%e (1,2) + (1,0),

%e (1,1) + (1,0) + (0,1).

%e First ten rows of triangle:

%e 0: 1

%e 1: 1 2

%e 2: 1 3 5

%e 3: 2 5 9 17

%e 4: 2 7 14 27 46

%e 5: 3 10 21 42 74 123

%e 6: 4 14 31 64 116 197 323

%e 7: 5 19 44 93 174 303 506 809

%e 8: 6 25 61 132 254 452 769 1251 1966

%e 9: 8 33 83 185 363 659 1141 1885 3006 4660

%t nmax = 10;

%t f[x_, y_] := Product[1 + x^n y^k, {n, 0, nmax}, {k, 0, nmax}]/2;

%t se = Series[f[x, y], {x, 0, nmax}, {y, 0, nmax}];

%t coes = CoefficientList[se, {x, y}];

%t t[n_ /; n >= 0, k_] /; 0 <= k <= n := coes[[n-k+1, k+1]];

%t T[n_, k_] := t[n+k, k];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 08 2021 *)

%o (Haskell) -- see link.

%Y T(n,0) = A000009(n);

%Y T(1,0) = A036469(0); T(n,1) = A036469(n) for n > 0.

%Y See A054242 for another version.

%Y Cf. A000009, A054225, A201376.

%Y T(n,n) = A219554(n). Row sums give: A219557. - _Alois P. Heinz_, Nov 22 2012

%K nonn,tabl

%O 0,3

%A _Reinhard Zumkeller_, Nov 30 2011

%E Entry revised by _N. J. A. Sloane_, Nov 30 2011