cp's OEIS Frontend

By analogy with ordinary partitions (A000041). The empty partition gives T(0,0)=1 by definition. A054225 and A201377 give partitions of pairs into sums of distinct pairs. 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.

Or, triangle T(n,k) of bipartite partitions of n objects, k of which are black.

Or, number of ways to factor p^(n-k)*q^k where p and q are distinct primes.

In the paper by F. C. Auluck: "On partitions of bipartite numbers", p.74, in the formula for fixed m there should be factor 1/m!. The correct asymptotic formula is p(m, n) ~ (sqrt(6*n)/Pi)^m * exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*m!*n). - Vaclav Kotesovec, Feb 01 2016

T(n,k)=T(n,k-n) is the number of multiset partitions of the multiset {1^k, 2^(n-k)}, see example link. - Joerg Arndt, Jan 01 2024

Let R be the ring of power series in two countably infinite sets of variables x_1,y_1,x_2,y_2,... that are invariant under the diagonal action (i.e, the group S of permutations of positive integers acts by w(x_i)=x_{w(i)} and w(y_i)=y_{w(i)}). Then T(n,k) is the dimension of the (n,k)-bigraded piece of R, i.e., the bihomogeneous power series of degree n in the x-variables and k in the y-variables that are S-invariant. - Jeremy L. Martin, Nov 27 2024

Number of ways to factor p^n * q^n where p and q are distinct primes.

First in a series of triangular arrays which comprise subsequences of A096443(n).

The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.

Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.

To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011

Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014

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.

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.

Also first prepending column of the 4-Zeckendorf array (see Ericksen and Anderson).

N. J. A. Sloane observed already the relation between Hofstadter G/H-like sequences H_k and k-Zeckendorf arrays in May 2003, at least for k = 3 (see formula section and history of A005374). First observation most probably by Diego Torres, Nov 2002, relating the Hofstadter G/H-like sequences H_k with the k-Zeckendorf arrays and Lamé sequences of order k (see comments in A005375 and A005376).

Also first prepending column of the 5-Zeckendorf array (see Ericksen and Anderson).

N. J. A. Sloane observed already the relation between Hofstadter G/H-like sequences H_k and k-Zeckendorf arrays in May 2003, at least for k = 3 (see formula section and history of A005374). First observation most probably by Diego Torres, Nov 2002, relating the Hofstadter G/H-like sequences H_k with the k-Zeckendorf arrays and Lamé sequences of order k (see comments in A005375 and A005376).