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A144630 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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The initial entries in each row form A000515. The second entries give A144631. The final entries are the squares (A000290).
Row sums are A144632. The penultimate entries in each row appear to be 4*A014105. - N. J. A. Sloane, Jan 20 2009
LINKS
Wikipedia, Hilbert matrix (gives inverse Hilbert matric explicitly).
EXAMPLE
The first three inverse Hilbert matrices are:
--------------
[ 1 ]
--------------
[4 -6 ]
[-6 12]
--------------
[ 9 -36 30 ]
[-36 192 -180]
[30 -180 180]
--------------
Triangle begins:
1,
12, 4,
180, 12, 9,
2800, 880, 40, 16,
44100, 46900, 4480, 40, 25,
698544, 1615824, 411264, 13104, 84, 36
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A047709 A002911 A038330 * A107670 A157782 A335949
KEYWORD
nonn,tabl
AUTHOR
Daniel McLaury and Ben Golub, Dec 23 2008
EXTENSIONS
More terms from R. J. Mathar and Klaus Brockhaus, Jan 21 2009
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)