0,4

The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1). From expansion of falling factorials (binomial transform of A005493)). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492 (From expansion of falling factorials.).

Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006

The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 08 2006

F. Ruskey, Combinatorial Generation, preprint, 2001.

F. Ruskey, Lexicographic Algorithms

For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)

A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1 - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006

(PARI) PARI code for printing k+1 terms of n-th row: f(n, k)=round(suminf(i=0, (i+k)^n/i!)/exp(1)) g(n, k)=for(k=0, k, print1(f(n, k), ", "))

Cf. A000110, A000110, A005493, A005494, A045379, A002522, A005491, A005492, A086659.

Adjacent sequences: A108084 A108085 A108086 this_sequence A108088 A108089 A108090

Sequence in context: A033184 A110488 A134379 this_sequence A123158 A133611 A010094

nonn,tabl

Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jun 05 2005