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.

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

Offset: 1

Author

Alois P. Heinz, Nov 26 2010

Keywords

• nonn
• tabl

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

Examples

			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,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..60, flattened
• Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210. - From _N. J. A. Sloane_, May 27 2012
• Eric Weisstein's World of Mathematics, Cycle Graph
• Eric Weisstein's World of Mathematics, Path Graph
• Wikipedia, Kirchhoff's theorem

Crossrefs

Columns k=1-11 give: A000012, A001542, A003690, A003753, A003733, A158880, A158898, A210812, A174001, A210813, A174089.

Rows n=1-2 give: A000027, A006235.

Main diagonal gives A252767.

Cf. A156308.

Programs

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);
# Crude Maple program from N. J. A. Sloane, May 27 2012:
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 *)

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 equals k times the resultant (R(k,x), U(n-1,(2 - x)/2)). - Peter Bala, May 01 2014 [Corrected by Pontus von Brömssen, Apr 08 2025]