cp's OEIS Frontend

Set partitions of the n-set into at most 6 parts; also restricted growth strings (RGS) with six letters s(1),s(2),...,s(6) where the first occurrence of s(j) precedes the first occurrence of s(k) for all j < k. - Joerg Arndt, Jul 06 2011

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

Density of regular language L over {1,2,3,4,5,6}^* (i.e., number of strings of length n in L) described by regular expression with c=6: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*) where Sum stands for union and Product for concatenation. - Nelma Moreira, Oct 10 2004

Word structures of length n using an N-ary alphabet are generated by taking M^n* the vector [(N 1's),0,0,0,...], leftmost column term = a(n+1). In the case of A056273, the vector = [1,1,1,1,1,1,0,0,0,...]. As the vector approaches all 1's, the leftmost column terms approach A000110, the Bell sequence. - Gary W. Adamson, Jun 23 2011

From Gary W. Adamson, Jul 06 2011: (Start)

Construct an infinite array of sequences representing word structures of length n using an N-ary alphabet as follows:

.

1, 1, 1, 1, 1, 1, 1, 1, ...; N=1, A000012

1, 2, 4, 8, 16, 32, 64, 128, ...; N=2, A000079

1, 2, 5, 14, 41, 122, 365, 1094, ...; N=3, A007051

1, 2, 5, 15, 51, 187, 715, 2795, ...; N=4, A007581

1, 2, 5, 15, 52, 202, 855, 3845, ...; N=5, A056272

1, 2, 5, 15, 52, 203, 876, 4111, ...; N=6, A056273

...

The sequences tend to A000110. Finite differences of columns reinterpreted as rows generate A008277 as a triangle: (1; 1,1; 1,3,1; 1,7,6,1; ...). (End)

We say that the sequence S is the exponential limit of the function f relative to the kernel K if and only if the exponential generating functions

egf(n) = Sum_{k=0..n} K(n, k)*f(x*(n-k)) generate a family of sequences

T(n) = k -> (k!/n!)*[x^k] egf(n) which converge to S. Convergence here means that for every fixed k the terms T(n)(k) differ from S(k) only for finitely many indices.

The paradigmatic example is to set f(x) = exp(x), K(n, k) = !k*binomial(n, k) (!n is the subfactorial of n) and obtain for S the Bell numbers. This example is set forth in A320955.

Let D(f)(x) represent the derivative of f(x) with respect to x and (D^(n))(f) the n-th derivative of f. Then the exponential limit of f is B(n)*((D^(n))(f))(0) where B(n) is the n-th Bell number: ExpLim(f) = f(0), (D(f))(0), 2*((D^(2))(f))(0), 5*((D^(3))(f))(0), 15*((D^(4))(f))(0), 52*((D^(5))(f))(0), ... Since exp is a fixed point of D and exp(0) = 1 we have the identity ExpLim(exp)[n] = B(n). Similarly ExpLim(sin)[n] = B(n)*mod(n,2)*(-1)^binomial(n,2).

If we set f = sec + tan and K(n, k) = !k*binomial(n, k) the exponential limit is this sequence, a(n).

See A320956 for definitions and comments.

See A320956 for definitions and comments.

The row sums of A320955 seen as a triangle are the partial sums of the antidiagonal sums of the triangle of the Stirling set numbers.

Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block m-j contains n-j (m in {0..n}, j in {0..m-1}). a(5) = 9: 12345, 1234|5, 123|4|5, 124|35, 12|3|4|5, 134|25, 13|24|5, 14|235, 1|2|3|4|5. - Alois P. Heinz, May 16 2023

See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.

The exponential limit of a function is defined in A320956. Applied to (-x)! Maple returns a sequence of sums of Zeta values, powers of Pi, powers of Euler's gamma, etc.. The sequence starts: 1, gamma, (1/3)*Pi^2 + 2* gamma^2, 10*Zeta(3) + (5/2)*Pi^2*gamma + 5*gamma^3, ... These sums, rounded to the nearest integer, give the sequence.