cp's OEIS Frontend

See A129065 for the M. Bruschi et al. reference.

See A129065 for the M. Bruschi et al. reference.

See A129065 for the M. Bruschi et al. reference.

Row sums are:

{1, 1, 3, 23, 329, 7545, 253195, 11692735, 710944785, 55043460305,

5286546264275}

Row sums are:

{1, 1, -1, -3, 15, 3, -387, 1827, 8001, -172935, 735399}

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002

If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009

a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011

a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011

a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

a(n-1) is the number of elements of the wreath product of S_n and S_2 with cycle partition equal to (2n), where S_n is the symmetric group of order n. - Josaphat Baolahy, Mar 12 2024

The row polynomials p(n,x) = Sum_{k=0..n} T(n,k)*x^k have the n integer zeros 2*A000217(j), j=0..n-1.

The row polynomials satisfy a three-term recurrence relation which qualify them as orthogonal polynomials w.r.t. some (as yet unknown) positive measure.

Column sequences (without leading zeros) give A000007, A010790(n-1)*(-1)^(n-1), A084915(n-1)*(-1)^(n-2), A130033 for m=0..3.

Apparently this is the triangle read by rows of Legendre-Stirling numbers of the first kind. See the Andrews-Gawronski-Littlejohn paper, table 2. The mirror version is the triangle A191936. - Omar E. Pol, Jan 10 2012

For v >= 1 the orthogonal polynomials p(n,v,x) have v integer zeros k*(k-1), k=1..v, for every n >= v. These zeros are from 2*A000217.

Coefficients of p(n,v=2,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.

The column sequences give A019590, A129464, A129465, A129466 for m=0,1,2,3.

p(n,v=2,x) has, for every n >= 2, simple zeros for integers x=0 and x=2. p(2,2,x) has therefore only integer zeros 0 and 2. det(V(n,2))=0 for every n >= 2.

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.