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A175243
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Array read by antidiagonals: total number of spanning trees R_n(m) of the complete prism K_m X C_n.
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2
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1, 2, 1, 3, 12, 3, 4, 75, 294, 16, 5, 384, 11664, 16384, 125, 6, 1805, 367500, 5647152, 1640250, 1296, 7, 8100, 10609215, 1528823808, 6291456000, 259200000, 16807, 8, 35287, 292626432, 380008339280, 18911429680500, 13556617751088, 59549251454
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OFFSET
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1,2
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LINKS
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FORMULA
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R_n(m) = n*2^(m-1)* (T(n,1+m/2)-1)^(m-1)/m, where T(n,x) are Chebyshev polynomials, A008310.
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
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EXAMPLE
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The array starts in row n=1 as:
1, 1, 3, 16, 125
2, 12, 294, 16384, 1640250
3, 75, 11664, 5647152, 6291456000
4, 384, 367500, 1528823808,
5, 1805, 10609215,
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MAPLE
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A175243 := proc(n, m) n*2^(m-1)/m*( orthopoly[T](n, 1+m/2)-1)^(m-1) ; end proc:
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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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