cp's OEIS Frontend

Hankel transform of n! (A000142(n)) and of A003319. - Paul Barry, Oct 07 2008

Hankel transform of A000255. - Paul Barry, Apr 22 2009

Monotonic magmas of size n, i.e., magmas with elements labeled 1..n where product(i,j) >= max(i,j). - Chad Brewbaker, Nov 03 2013

Also called the bouncing factorial function. - Alexander Goebel, Apr 08 2020

For v >= 1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k = 1..v, for every n >= v.

Coefficients of p(n,v=1,x) (in the quoted Bruschi, et al., paper p(nu, n)(x) of eqs. (4) and (8a),(8b)) in increasing powers of x.

The v-family p(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix V=V(M,v) with entries V_{m,n} given by v*(v-1) - (m-1)^2 - (v-m)^2 if n=m, m=1,...,M; (m-1)^2 if n=m-1, m=2,...,M; (v-m)^2 if n=m+1, m=1..M-1 and 0 else. p(n,v,x) := det(x*I_n - V(n,v) with the n-dimensional unit matrix I_n.

p(n,v=1,x) has, for every n >= 1, a zero for x=0, i.e., det(V(n,1))=0 for every n >= 1. This is obvious.

The column sequences give A000007, A010790, A129460, A129461 for m=0,1,2,3.

Column 2 in triangle A009963.

a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.

Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017

Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022

The ZG1 matrix coefficients are defined by ZG1[2m-1,1] = 2*zeta(2m-1) for m = 2, 3, .. , and the recurrence relation ZG1[2m-1,n] = (ZG1[2m-3,n-1] - (n-1)^2*ZG1[2m-1,n-1])/(n*(n-1)) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= (m-1). As usual zeta(m) is the Riemann zeta function. For the ZG2 matrix, the even counterpart of the ZG1 matrix, see A008955.

These two formulas enable us to determine the values of the ZG1[2*m-1,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, ZG1[1,1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.

The coefficients in the columns of the ZG1 matrix, for m >= 1 and n >= 2, can be generated with GFZ(z;n) = (hg(n)*CFN1(z;n)*GFZ(z;n=1) + ZETA(z;n))/pg(n) with pg(n) = 6*(n-1)!* (n)!*A160476(n) and hg(n) = 6*A160476(n). For the CFN1(z;n) and the ZETA(z;n) polynomials see A160474.

The column sums cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity), for n >= 2, of the ZG1 matrix can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take ZGx[2*m-1,n] = 2, for m >= 1, and ZGx[ -1,n] = ZG1[ -1,n] and assume that the recurrence relation remains the same we find that the column sums of this new matrix converge to the same values as the original cs(n).

The ZG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.

The ZG1 matrix is related to the ZS1 matrix, see A160474 and the formulas below.

Also the number of vertex covers and independent vertex sets of the n X n bishop graph.

a(n) is the number of reduced Stable Marriage Problem instances of order n. In the SMP, relabeling men or women has no effect on the number of stable matchings. So the men and women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and to similarly normalize the order of woman #1's rankings, except for her ranking of man #1. This reduces the number of possible instances by a factor of n!(n-1)! (A010790 with shifted offset), from (n!)^(2n) (A185141) to a(n). This reduction is directly analogous to the identical reduction from latin squares (A002860) to reduced latin squares (A000315), and can be directly applied to the Latin Stable Marriage Problem (A351413). As with reduced latin squares, some further reduction is possible analogous to row/column reduced latin squares (A123234).

It is tempting to aim for a reduction of (n!)^2 by simultaneously normalizing all of man #1 and woman #1's preferences, but that isn't possible unless man #1 and woman #1 happen to be mutual first choices.

Applying this reduction to A344669 reduces A344669(2) and A344669(4) to 1, demonstrating that these maximal instances arising in A005154 are unique up to participant relabeling. It raises the question of which other values of n make A344669(n) reducible to 1.

Similar to, but different from, superfactorial Pascal triangle A009963.

A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.

From Natalia L. Skirrow, Apr 13 2025 (Start)

Denoting this sequence as the superbinomial sb(n,k), the hook length formula for a j X k rectangular Young tableau states the number of configurations of j*k distinct numbers such that each row and column is strictly increasing is (j*k)!/sb(j+k,j), ie. 1/sb(j+k,j) is the probability that a random permutation is a Young tableau.

Meanwhile, if the numbers are placed into the array with repetition, but the columns are still strictly increasing, there are c(n,j,k) = sb(n+1,j+k)/(sb(n+1-j,k)*sb(n+1-k,j)) configurations.

If the strict criterion is relaxed to monotonic, this becomes C(n,j,k) = sb(n-1+j+k,j+k)/(sb(n-1+j,j)*sb(n-1+k,k)).

By proposition 13.2(i) of Stanley's PhD thesis, for fixed j,k, c(n,j,k) and C(n,j,k) are polynomials in n of degree j*k, and c(n,j,k) = (-1)^(j*k)*C(-n,j,k).

For example, c(n,1,k)=(n choose k) and C(n,1,k)=(n+k-1 choose k), while c(n,2,k) = N(n,k+1) and C(n,2,k) = N(n+k,k+1), so the binomial coefficients and Narayana numbers N=A001263 obey the dualities (under continuation as polynomials) (n choose k) = (-1)^k*(k-1-n choose k) and N(n,k) = N(k-1-n,k).

(End)

For v>=1 the orthogonal polynomials pt(n,v,x) have v integer zeros k*(k+1), k=1..v, for every n>=v and some other n-v zeros. The integer zeros are from 2*A000217.

The v-family pt(n,v,x) consists of characteristic polynomials of the tridiagonal M x M matrix Vt=Vt(M,v) with entries Vt_{m,n} given by 2*m*(v+1-m) if n=m, m=1,...,M; -m*(v+1-m) if n=m-1, m=2,...,M; -m*(v+1-m) if n=m+1, m=1..M-1 and 0 else. pt(n,v,x):=det(x*I_n-Vt(n,v)) with the n dimensional unit matrix I_n.

pt(n,v=1,x) has, for every n>=1, among its n zeros one for x=2. pt(1,1,x) has therefore only the integer zeros 2. det(Vt(1,1))=2.

The column sequences give [1,-2,0,0,0,...], A010790(n-1)*(-1)^(n-1), A130185, A130186 for m=0,1,2,3.

Coefficients of pt(n,v=1,x) (in the quoted Bruschi et al. paper {\tilde p}^{(\nu)}_n(x) of eqs. (20) and (24a),(24b)) in increasing powers of x.