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a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.

a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 5*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.

a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.

a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x-6*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.

T(n,m) = R_n^m(a=0, b=7) in the notation of the given 1962 reference.

T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x-7*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

From Petros Hadjicostas, Jun 07 2020: (Start)

For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962).

They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.

With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m).

We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.

For the current array, T(n,m) = R_n^m(a=0, b=7) but with no zero row or column. (End)

T(n,m)= R_n^m(a=0, b=8) in the notation of the given 1962 reference.

T(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 8*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

From Petros Hadjicostas, Jun 07 2020: (Start)

For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962). Such numbers are related to the work of Nörlund (1924).

They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.

With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m).

We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.

For the current array, T(n,m) = R_n^m(a=0, b=8) but with no zero row or column. (End)

The Catalan tree is defined as follows: the root is labeled 1 and each vertex labeled i has i+1 children labeled 1,2,...,i+1. The weight of a vertex v is the product of all labels on the path from the root to v. Row n contains n+1 terms. Row sums and column 1 yield the double factorials (A001147). T(n,n+1)=(n+1)!, T(n,n)=n(n+1)!/2 (A001286; Lah numbers).

This table counts permutations of the multiset {1,1,2,2,...,n,n} satisfying the condition "the first appearance of i + 1 follows the first appearance of i" by the position of the first appearance of n. Specifically, T(n+1,k) is the number of such permutations for which n first occurs in position 2n+1-k. For example, with n=2 and k=1, T(3,1)=6 counts 121323, 121332, 122313, 122331, 112323, 112332. - David Callan, Nov 29 2007

T(n+1,k) is also the number of rooted complete binary forests with n labeled leaves and k labeled roots. This follows by comparing exponential generating functions; see Example 5.2.6 and Proposition 5.1.3 of Stanley's "Enumerative Combinatorics 2." - Timothy Y. Chow, Mar 28 2017

Also the Bell transform of the triple factorial numbers A007559 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

The W. Lang link in A036039 explicitly gives the first several cycle index polynomials for the symmetric group S_n, or the partition polynomials for the refined Stirling numbers of the first kind. In line with the discussion in the Fecko link, null the indeterminates with odd indices, divide the 2n-th partition polynomial by the double factorial of odd numbers given in A001147, and re-index. The sum of the resulting row coefficients are also equal to A001147.

Row sums are zero if n>0.

Building the symmetric form A(n,m)+A(n,n-m) as here is equivalent to tabulation of the coefficients of a polynomial p_n(x) of order n associated with A(.,.) plus its reverse: t(n,m) = [x^m] ( p_n(x)+x^n*p_n(1/x)), here with p_n(x)=product(x-(2i-1)). Note that the product of the polynomial p_n(x) = sum_{m>=0} A(n,m)*x^m and the polynomial p'n(x)= sum{m>=0} A(n,n-m)*x^m is given in terms of (terminating) generating function by a convolution, related to the reversal of the sense of the second index in the A(n,m). So the fact that one can obtain A(n,n-m) by using the reverse polynomial p'_n(x) = x^n/p_n(x) is by no means special to this sequence here. The consequence that A(n,m)+A(n,n-m) defines a left-right symmetric row is then obvious.