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A144630
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Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the entries in the lower right k X k submatrix of the n X n inverse Hilbert matrix.
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3
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1, 12, 4, 180, 12, 9, 2800, 880, 40, 16, 44100, 46900, 4480, 40, 25, 698544, 1615824, 411264, 13104, 84, 36, 11099088, 45094896, 23653476, 2268756, 36036, 84, 49, 176679360, 1115345088, 1017615456, 207193536, 9660816, 79776, 144, 64
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OFFSET
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1,2
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COMMENTS
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The initial entries in each row form A000515. The second entries give A144631. The final entries are the squares (A000290).
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LINKS
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EXAMPLE
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The first three inverse Hilbert matrices are:
--------------
[ 1 ]
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[4 -6 ]
[-6 12]
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[ 9 -36 30 ]
[-36 192 -180]
[30 -180 180]
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Triangle begins:
1,
12, 4,
180, 12, 9,
2800, 880, 40, 16,
44100, 46900, 4480, 40, 25,
698544, 1615824, 411264, 13104, 84, 36
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MAPLE
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invH := proc(n, i, j) (-1)^(i+j)*(i+j-1)*binomial(n+i-1, n-j)*binomial(n+j-1, n-i)* (binomial(i+j-2, i-1))^2 ; end: A144630 := proc(n, k) local T, i, j ; T := 0 ; for i from n-k+1 to n do for j from n-k+1 to n do T := T+invH(n, i, j) ; od; od; RETURN(T) ; end: for n from 1 to 10 do for k from 1 to n do printf("%a, ", A144630(n, k)) : od: od: # R. J. Mathar, Jan 21 2009
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MATHEMATICA
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inverseHilbert[n_, i_, j_] := (-1)^(i+j)*(i+j-1) * Binomial[n+i-1, n-j] * Binomial[n+j-1, n-i] * Binomial[i+j-2, i-1]^2; inverseHilbert[n_, k_] := Table[ inverseHilbert[n, i, j], {i, n-k+1, n}, {j, n-k+1, n}]; t[n_, k_] := Tr[ Flatten[ inverseHilbert[n, k]]]; Flatten[ Table[t[n, k], {n, 1, 8}, {k, 1, n}]] (* Jean-François Alcover, Jul 16 2012 *)
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PROG
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(MATLAB) invhilb(1), invhilb(2), invhilb(3), etc.
(Magma) &cat[ [ &+[I[i][j]: i, j in [k..n] ]: k in [n..1 by -1] ] where I:=H^-1 where H:=Matrix(Rationals(), n, n, [ < i, j, 1/(i+j-1) >: i, j in [1..n] ] ): n in [1..8] ]; // Klaus Brockhaus, Jan 21 2009
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CROSSREFS
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KEYWORD
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AUTHOR
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Daniel McLaury and Ben Golub, Dec 23 2008
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EXTENSIONS
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STATUS
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approved
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