|
|
A173958
|
|
Number A(n,k) of spanning trees in C_k X P_n; square array A(n,k), n>=1, k>=1, read by antidiagonals.
|
|
13
|
|
|
1, 2, 1, 3, 12, 1, 4, 75, 70, 1, 5, 384, 1728, 408, 1, 6, 1805, 31500, 39675, 2378, 1, 7, 8100, 508805, 2558976, 910803, 13860, 1, 8, 35287, 7741440, 140503005, 207746836, 20908800, 80782, 1, 9, 150528, 113742727, 7138643400, 38720000000, 16864848000, 479991603, 470832, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Every row and every column of the array is a divisibility sequence, i.e., the terms satisfy the property that if n divides m then a(n) divides a(m) provided a(n) > 0. This follows from the representation of the elements of the array as a resultant. - Peter Bala, May 01 2014
|
|
LINKS
|
|
|
FORMULA
|
A(n,k) = m*Prod(Prod( 4*sin(h*Pi/m)^2+4*sin(k*Pi/(2*n))^2, h=1..m-1), k=1..n-1) [Kreweras]. - From N. J. A. Sloane, May 27 2012
Let T(n,x) and U(n,x) denote the Chebyshev polynomials of the first and second kind respectively. Let R(n,x) = 2*( T(n,(x + 2)/2) - 1 )/x (the row polynomials of A156308). Then the (n,k)-th element of the array = resultant (R(k,x), U(n-1,(2 - x)/2). - Peter Bala, May 01 2014
|
|
EXAMPLE
|
Square array A(n,k) begins:
1, 2, 3, 4, 5, ...
1, 12, 75, 384, 1805, ...
1, 70, 1728, 31500, 508805, ...
1, 408, 39675, 2558976, 140503005, ...
1, 2378, 910803, 207746836, 38720000000, ...
|
|
MAPLE
|
with(LinearAlgebra):
A:= proc(n, m) local M, i, j;
if m=1 then 1 else
M:= Matrix(n*m, shape=symmetric);
for i to n do
for j to m-1 do M[m*(i-1)+j, m*(i-1)+j+1]:=-1 od;
M[m*(i-1)+1, m*i]:= M[m*(i-1)+1, m*i]-1
od;
for i to n-1 do
for j to m do M[m*(i-1)+j, m*i+j]:=-1 od
od;
for i to n*m do
M[i, i]:= -add(M[i, j], j=1..n*m)
od;
Determinant(DeleteColumn(DeleteRow(M, 1), 1))
fi
end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..9);
Digits:=200;
T:=(m, n)->round(Re(evalf(simplify(expand(
m*mul(mul( 4*sin(h*Pi/m)^2+4*sin(k*Pi/(2*n))^2, h=1..m-1), k=1..n-1))))));
# Alternative program using the resultant:
for n from 1 to 10 do seq(k*resultant(simplify((2*(ChebyshevT(k, (x + 2)/2) - 1))/x), simplify(ChebyshevU(n-1, 1 - x/2)), x), k = 1 .. 10) end do; # Peter Bala, May 01 2014
|
|
MATHEMATICA
|
t[m_, n_] := m*Product[Product[4*Sin[h*Pi/m]^2 + 4*Sin[k*Pi/(2*n)]^2, {h, 1, m-1}], {k, 1, n-1}]; Table[t[m, n-m+1] // Round, {n, 1, 9}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 05 2013, after N. J. A. Sloane *)
|
|
CROSSREFS
|
Columns k=1-11 give: A000012, A001542, A003690, A003753, A003733, A158880, A158898, A210812, A174001, A210813, A174089.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|