0,5

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructable from its row and column sums. [Brewbaker]

B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k } [Launois]

T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)

Chad Brewbaker, Enumerating (0, 1) Matrices Uniquely Reconstructable From Their Row and Column Sum Vectors

M. Kaneko, Poly-Bernoulli numbers

S. Launois, Combinatorics of H-primes in quantum matrices

pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ]

E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].

T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 04 2004

1,1,1,1,1,1,

1,2,4,8,16,32,

1,4,14,46,146,454,

1,8,46,230,1066,4718,

1,16,146,1066,6902,41506,

1,32,454,4718,41506,329462,

(PARI) T(n, k)=sum(j=0, n, (j+1)^k*sum(i=0, j, (-1)^(n+j-i)*binomial(j, i)*(j-i)^n))

Rows 0-4 are A000012, A000079, A027649, A027650, A027651. Main diagonal is A048163. Antidiagonal sums are in A098830. Cf. A019538.

Sequence in context: A062715 A100631 A064298 this_sequence A117401 A144324 A034372

Adjacent sequences: A099591 A099592 A099593 this_sequence A099595 A099596 A099597

nonn,tabl

Ralf Stephan, Oct 27 2004