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A refinement of A135278, up the sign these are the coefficients appearing in the expansion of power-sum symmetric functions in terms of elementary or homogeneous symmetric functions.

Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.

Row sums of A095705. - David Wasserman, Feb 22 2008

From Ludovic Schwob, Sep 23 2023: (Start)

a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:

[1 1 1] [1 2] [2 1] [3]

.

[1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]

[1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]

.

[1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]

[1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]

[1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]

.

[0 1 0] [0 1 0] [0 0 1] [0 0 1]

[1 0 0] [0 0 1] [1 0 0] [0 1 0]

[0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)

Also, partitions such that a set of k equal terms are labeled 1 through k and can appear in any order. For example, the partition 3+2+2+2+1+1+1+1 of 13 appears 1!*3!*4!=144 times because there are 1! ways to order the one "3," 3! ways to order the three "2"s, ... - Christian G. Bower, Jan 17 2006

This array was inspired by R. H. Hardin's recurrences for the columns of array A212855. Rows k=1 to k=5 are due to him, while the remaining rows were computed by Alois P. Heinz.

Row n has length A000041(n) + 1, i.e., one more than the number of partitions of n.

Let R(m,n) := R(m,n,t=0) = A212855(m,n) for m,n >= 1, where R(m,n,t) = LHS of Eq. (6) of Abramson and Promislow (1978, p. 248).

Let P_n be the set of all lists a = (a_1, a_2,..., a_n) of integers a_i >= 0, i = 1,..., n such that 1*a_1 + 2*a_2 + ... + n*a_n = n; i.e., P_n is the set all integer partitions of n. (We use a different notation for partitions than the one in the name of T(n,k).) Then |P_n| = A000041(n) for n >= 0.

We have R(m,n) = A212855(m,n) = Sum_{a in P_n} (-1)^(n - Sum_{j=1..n} a_j) * (a_1 + a_2 + ... + a_n)!/(a_1! * a_2! * ... * a_n!) * (n! / ((1!)^a_1 * (2!)^a_2 * ... * (n!)^a_n))^m.

The recurrence of R. H. Hardin for column n of array A212855 is Sum_{s = 0..|P_n|} (-1)^s * T(n,s) * R(m-s,n) = 0 for n >= 1 and m >= |P_n| + 1.

The above recurrence is correct for all n >= 1, but it is not always a minimal one. For example, it seems to be the minimal one for n = 1,...,6, but not for n = 7 (see A212854). It seems to be minimal whenever every two different partitions of n give different multinomial coefficients.

For n = 7, the partitions (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (0, 2, 1, 0, 0, 0, 0) (i.e., 2 + 2 + 3) and (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (3, 0, 0, 1, 0, 0, 0) (i.e., 1 + 1 + 1 + 4) give the same multinomial coefficient: 210 = 7!/(2!2!3!) = 7!/(1!1!1!4!). Hence, to find the minimal recurrence for n = 7, we count 210 only once in the set of multinomial coefficients: 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040. Then the absolute value of the coefficient of a(n-1) in the minimal recurrence is the sum of these multinomial coefficients (i.e., 11271); the absolute value of the coefficient of a(n-2) in the minimal recurrence is the sum of products of every two of them (i.e., 46169368), and so on.

Looking at the multinomial coefficients of the integer partitions of n = 8, 9, 10 on pp. 831-832 of Abramowitz and Stegun (1964), we see that, even in these cases, the above recurrence is not the minimal one. The number of distinct multinomial coefficients among the integer partitions of n is given by A070289.

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

Equals sums of the squares of terms in rows of the triangle of multinomial coefficients (A036038).

Ignoring initial term, equals the logarithmic derivative of A183241; A183241 is conjectured to consist entirely of integers.

More generally, let {M(n,k), n>=0} be the sums of the k-th powers of the multinomial coefficients where k>=0 (see table A183610), then:

Sum_{n>=0} M(n,k)*x^n/n!^k = Product_{n>=1} 1/(1-x^n/n!^k).

A generalization of the triangle of Stirling numbers of the second kind, these are the coefficients appearing in the expansion of (x_1 + x_2 + x_3 + ...)^n in terms of augmented monomial symmetric functions. They also appear in Faa di Bruno's formula.