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Conjecture: (i) a(n) > 0 for all n > 1. Also, for any integer n > 4, p(k)*q(n-k) - 1 is prime for some 0 < k < n/2.

(ii) If n > 9, then prime(k)*p(n-k) + 1 is prime for some 0 < k < n. If n > 2, then prime(k)*q(n-k) - 1 is prime for some 0 < k < n, and also prime(k)*q(n-k) + 1 is prime for some 0 < k < n.

(iii) If n > 11, then prime(k) + p(n-k) is prime for some 0 < k < n. If n > 4, then prime(k) + q(n-k) is prime for some 0 < k < n, and also prime(k)^2 + q(n-k)^2 is prime for some 0 < k < n.

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

Conjecture: (i) a(n) > 0 for all n > 727.

(ii) For the strict partition function q(.) (cf. A000009), any n > 93 can be written as k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and q(p-1) - 1 are both prime.

(iii) If n > 75 is not equal to 391, then n can be written as k + m with k > 0 and m > 0 such that f(k,m) - 1, f(k,m) + 1 and q(f(k,m)) + 1 are all prime, where f(k,m) = phi(k) + phi(m)/2.

Part (i) of the conjecture implies that there are infinitely many primes p with P(p-1) prime.

Conjecture: all divisors of all terms of row n are also all parts of all partitions of n.

The conjecture gives a correspondence between divisors and partitions (see example).

It is conjectured that every section of the set of partitions of n has essentially the same correspondence. For more information see A336811.

Coefficients are listed in Abramowitz and Stegun order (A036036).

The formula for the inversion of the power series y = F(x) = x*G(x) = x*(1 + Sum_{k>=1} g[k]*(x^k)) is obtained as a corollary of Lagrange's inversion theorem. The result is F^{(-1)}(y)= Sum_{n>=1} P(n-1)*y^n, where P(n):=sum over partitions of n of a(n,k)* G[k], with G[k]:=g[1]^e(k,1)*g[2]^e(k,2)*...*g[n]^e(k,n) if the k-th partition of n, in Abramowitz-Stegun order(see the given ref, pp. 831-2), is [1^e(k,1),2^e(k,2),...,n^e(k,n)], for k=1..p(n):= A000041(n) (partition numbers).

The sequence of row lengths is A000041(n) (partition numbers).

The signs are given by (-1)^m(n,k), with the number of parts m(n,k) = Sum_{j=1..n} e(k,j) of the k-th partition of n. For m(n,k) see A036043.

The proof that the unsigned row sums give Schroeder's little numbers A001003(n) results from their formula ((d^(n-1)/dx^(n-1)) ((1-x)/(1-2*x))^n)/n!|_{x=0}, n >= 1. This formula for A001003 can be proved starting with the compositional inverse of the g.f. of A001003 (which is given there in a comment) and using Lagrange's inversion theorem to recover the original sequence A001003.

For alternate formulations and relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [Tom Copeland, Sep 29 2008]

The coefficients of the row polynomials P(n) with monomials in lexicographically descending order e.g. P(6) = -1*g[6] + 8*g[5]*g[1] + 8*g[4]*g[2] - 36*g[4]*g[1]^2 + 4*g[3]^2 - 72*g[3]*g[2]*g[1] - 12*g[2]^3 + 120*g[3]*g[1]^3 + 180*g[2]^2*g[1]^2 - 330*g[2]*g[1]^4 + 132*g[1]^6 are given in A304462. [Herbert Eberle, Aug 16 2018]

Assume we have N <= n variables x_1, x_2, ..., x_N, and let S_n^{(N)} = x_1^n + x_2^n + ... + x_N^n be the n-th power sum of these variables. Following Gould (1999, p. 135), define the elementary symmetric polynomials e_1^{(N)}, e^2^{(N)}, ..., e_N^{(N)}, where e_j^{(N)} is the sum of all products of x_1, x_2, ..., x_N taken j at a time.

Since N <= n, without loss of generality, we may assume we have the extra variables x_{N+1}, ..., x_n, define e_1^{(n)}, e_2^{(n)}, ..., e_N^{(n)}, e_{N+1}^{(n)}, ..., e_n^{(n)}, and then set x_{N+1} = x_{N+2} = ... = x_n = 0. In such a case, we get e_j^{(N)} = e_j^{(n)} for j = 1..N, and e_j^{(n)} = 0 for j = (N+1)..n. Thus, we may drop the superscripts N and n on power sum and on the elementary symmetric polynomials and write S_n, and e_1, e_2, ..., e_n with the understanding that e_{N+1} = ... = e_n = 0. (See also the comments for A115131.)

The numbers T(n,k) are the coefficients of the power sum expansion in terms of the elementary symmetric polynomials, which is the Girard-Waring formula. That is, S_n = Sum(c(t_1,...,t_n) * e_1^t_1 * e_2^t_2 *...* e_n^t_n), summed over integer partitions of n, t_1 + 2*t_2 + ... + n*t_n = n with t_i >= 0 for i = 1..n. Here c(t_1,t_2,...,t_n) = n * (-1)^(t_2 + t_4 + ... + t_{2*floor(n/2)}) * (t_1 + ... + t_n - 1)!/(t_1!*...*t_n!).

Given a partition (t_1,...,t_n) of n (as defined above), we may use only those j's in {1,...,n} for which t_j > 0 and write the partition in the usual notation by repeating each such j t_j times (and place these j's in a non-descending order). E.g., the partition 1*3 + 2*1 + 3*0 + 4*0 + 5*0 of 5 can be written as [1,1,1,2]. Similarly, the partition 1*5 + 2*0 + 3*0 + 4*0 + 5*0 of 5 can be written as [1,1,1,1,1].

Instead of using the Abramowitz-Stegun order of partitions (as it is done in A115131), we use the usual notation for partitions (in terms of the positive integers that add up to n) and order them in lexicographic order. Unfortunately, this order of partitions does not correspond to any of the orders in the web link below.

If we let a(m) be the m-th term of the array, read as sequence, then

a(1) = T(1,1) = c([1]) = c(1),

a(2) = T(2,1) = c([1,1]) = c(2,0) with sign(T(2,1)) = (-1)^0 = 1,

a(3) = T(2,2) = c([2]) = c(0,1) with sign(T(2,2)) = (-1)^1 = -1,

a(4) = T(3,1) = c([1,1,1]) = c(3,0,0) with sign(T(3,1)) = (-1)^0 = 1,

a(5) = T(3,2) = c([1,2]) = c(1,1,0) with sign(T(4,1)) = (-1)^1 = -1,

a(6) = T(3,3) = c([3]) = c(0,0,1) with sign(T(3,3)) = (-1)^0 = 1,

a(7) = T(4,1) = c([1,1,1,1]) = c(4,0,0,0) with sign(T(4,1)) = (-1)^(0+0) = 1,

a(8) = T(4,2) = c([1,1,2]) = c(2,1,0,0) with sign(T(4,2)) = (-1)^(1+0) = -1,

a(9) = T(4,3) = c([1,3]) = c(1,0,1,0) with sign(T(4,3)) = (-1)^(0+0) = 1,

a(10) = T(4,4) = c([2,2]) = c(0,2,0,0) with sign(T(4,4)) = (-1)^(2+0) = 1,

a(11) = T(4,5) = c([4]) = c(0,0,0,1) with sign(T(4,5)) = (-1)^(0+1) = -1,

a(12) = T(5,1) = c([1,1,1,1,1]) = c(5,0,0,0,0),

a(13) = T(5,2) = c([1,1,1,2]) = c(3,1,0,0,0),

a(14) = T(5,3) = c([1,1,3]) = c(2,0,1,0,0),

a(15) = T(5,4) = c([1,2,2]) = c(1,2,0,0,0),

a(16) = T(5,5) = c([1,4]) = c(1,0,0,1,0),

a(17) = T(5,6) = c([2,3]) = c(0,1,1,0,0),

a(18) = T(5,7) = c([5]) = c(0,0,0,0,1), ...

(ascending ordered compositions in lexicographic order).

T(n,k) is also the number of divisors of A336811(n,k).

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.