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%I #18 Jun 06 2023 10:50:08
%S 1,2,1,3,12,3,4,75,294,16,5,384,11664,16384,125,6,1805,367500,5647152,
%T 1640250,1296,7,8100,10609215,1528823808,6291456000,259200000,16807,8,
%U 35287,292626432,380008339280,18911429680500,13556617751088,59549251454
%N Array read by antidiagonals: total number of spanning trees R_n(m) of the complete prism K_m X C_n.
%H F. T. Boesch and H. Prodinger, <a href="http://dx.doi.org/10.1007/BF01788093">Spanning tree formulas and Chebyshev polynomials</a>, Graphs Combinat. 2 (1986) 191-200.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F R_n(m) = n*2^(m-1)* (T(n,1+m/2)-1)^(m-1)/m, where T(n,x) are Chebyshev polynomials, A008310.
%F Each column of the array is a linear divisibility sequence. Conjecturally, the k-th column satisfies a linear recurrence of order 4*k - 2. - _Peter Bala_, May 04 2014
%e The array starts in row n=1 as:
%e 1, 1, 3, 16, 125
%e 2, 12, 294, 16384, 1640250
%e 3, 75, 11664, 5647152, 6291456000
%e 4, 384, 367500, 1528823808,
%e 5, 1805, 10609215,
%p A175243 := proc(n,m) n*2^(m-1)/m*( orthopoly[T](n,1+m/2)-1)^(m-1) ; end proc:
%p for d from 2 to 10 do for m from 1 to d-1 do n := d-m ; printf("%d,",A175243(n,m)) ; end do: end do:
%t r[n_, m_] := n*2^(m-1)*(ChebyshevT[n, 1+m/2]-1)^(m-1)/m; Table[r[n-m, m], {n, 2, 9}, {m, 1, n-1}] // Flatten (* _Jean-François Alcover_, Jan 10 2014 *)
%Y Cf. A006235 (column 2), A000272, A212798 (column 3).
%K easy,nonn,tabl
%O 1,2
%A _R. J. Mathar_, Mar 13 2010
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