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.

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

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

Klaus Brockhaus, Table of n, a(n) for n=1..1830 (rows 1 - 60)

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

Cf. A000290, A000515, A144631.

Sequence in context: A047709 A002911 A038330 * A107670 A157782 A335949

Adjacent sequences: A144627 A144628 A144629 * A144631 A144632 A144633

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