This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 12
 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 Alois P. Heinz, Antidiagonals n = 1..45, flattened Germain Kreweras, Complexite et circuits Euleriens 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 FORMULA 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); # 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 *) 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. Sequence in context: A187111 A122050 A081323 * A175243 A168217 A329025 Adjacent sequences:  A173955 A173956 A173957 * A173959 A173960 A173961 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Nov 26 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)