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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (njas(AT)research.att.com), Jan 20 2009
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REFERENCES
| Wikipedia, Hilbert Matrix (gived inverse Hilbert matric explicitly).
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LINKS
| Klaus Brockhaus, Table of n, a(n) for n=1..1830 (rows 1 - 60)
Wikipedia, Hilbert matrix [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 21 2009]
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EXAMPLE
| The first three inverse Hilbert matrices are:
--------------
[ 1 ]
--------------
[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
| 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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2009]
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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] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 21 2009]
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CROSSREFS
| Cf. A000290, A000515, A144631.
Sequence in context: A047709 A002911 A038330 * A107670 A157782 A002679
Adjacent sequences: A144627 A144628 A144629 * A144631 A144632 A144633
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KEYWORD
| nonn,tabl
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AUTHOR
| Daniel McLaury and Ben Golub, Dec 23 2008
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 21 2009
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