A201376 - OEIS

%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