cp's OEIS Frontend

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

Triangular array distributing the Bell numbers (A000110). The value associated with each partition is the product of A074664(k) for each part of size k, times the number of compositions associated with the partition (A048996 & A072881). The value for T(n,k) is the total of these values for each partition of n into k parts.

Calculating the appropriate weights can be done by "working backward". Suppose for example we know the weights for 1 through 6 and desire the weight for the partitions of seven: Substitute the weights for each partition value and multiply. For example, 7 = 4+3 so f([4,3]) = 6*2 = 12; adjusting for the number of permutations of [4,3] we now have 2*12 = 24. Continuing in this manner for each partition of seven and summing to 451 we now know all of the values except that associated with the partition [7] which must be 877 - 451 = 426.

From Mike Zabrocki: (Start)

Every set partition can be uniquely split into "atomic" set partitions or is itself already atomic.

{{1},{2},{3}} = {{1}}|{{1}}|{{1}}

{{1},{23}} = {{1}}|{{12}}

{{12},{3}} = {{12}}|{{1}}

{{13},{2}} is already atomic

{{123}} is already atomic

where this operation | is defined as {A1,...,Ar}|{B1,...,Bs} = {A1,...,Ar,B1+n,...,Bs+n}

where Bi+n = {bi1+n,bi2+n,...,bik+n} if Bi = {bi1,bi2,...,bik} and n = |A1|+|A2|+...+|Ar|. (End)

Subtriangle (n >= 1 and 1 <= k <= n) of triangle given by [0,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 03 2007

From Peter Bala, Aug 05 2014: (Start)

Let B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + ... denote the o.g.f. for the Bell numbers A000110. Let f(x) = (B(x) - 1)/(x*B(x)) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + ..., the o.g.f. for the first column of this array. Then this array appears to be the Riordan array (f(x), x*f(x)).

If true, this gives the o.g.f. of the array as (B(x) - 1)/( x*(t + (1 - t)*B(x)) ) = 1 + (1 + t)*x + (2 + 2*t + t^2)*x^2 + ... and also the hockey-stick recurrence: T(n+1,k+1) = T(n,k) + T(n-1,k) + 2*T(n-2,k) + 6*T(n-3,k) + 22*T(n-4,k) + ..., n,k >= 1. (End)

Row n has A000041(n) terms.

Consider the five partitions of the number 4:

4 3+1 2+2 2+1+1 and 1+1+1+1

rewriting as 4000 3100 2200 2110 and 1111

then a(n) counts the ways that the zeros can be permuted:

6,2,2,1,1

agreeing with the factorial of the difference between A036042 and A036043.

Since A049085 is a resortment of A036043 both A179972 and A179974 have row sums equal to A179973.

The Data section here is longer than usual. Do not shorten it! - N. J. A. Sloane, Jan 10 2019

The length of row n is A000041(n), the number of partitions of n.

Partitions follow the Abramowitz-Stegun (A-St) order (see the link).

The row sums give A001906(n) = Fibonacci(2*n).

The triangle T(n, m) obtained by summing in row n the entries of the columns k with identical part number m is A078812(n, m) = binomial(n+m-1, 2*m-1) (with offsets n >= 1, m = 1..n). The array of the number of parts m = m(n,k) = A036043(n, k) in A-St order.

This array is the elementwise product of the array A048996, the composition numbers, and A118851, the products of the parts of partitions, both arrays are in A-St order.

Therefore a(n, k) is the sum of the number of products of the block lengths of all the A048996(n, k) set partition of [n] := {1,2, ..., n} with m = m(n, k) blocks consisting of consecutive numbers corresponding to the k-th partition of n in A-St order. Because the block structure depends only on the exponents (signature) of the underlying partition this leads to the product of the two array entries. Equivalently, one can consider compositions. Then a(n, k) gives the sum of the products of the parts of all A048996(n, k) compositions originating from the k-th partition of n.

This array is the result of an attempt to understand the comment of Kevin Long, May 11 2018, on A001906.

This array is similar to A085643 but some pairs of numbers like (27, 36), (72,48), (54,144), ... are there swapped.

First differs from A118457 at a(53) = 4, A118457(53) = 2.

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.