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

Author

Daniel McLaury and Ben Golub, Dec 23 2008

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

• Klaus Brockhaus, Table of n, a(n) for n=1..1830 (rows 1 - 60)
• Wikipedia, Hilbert matrix (gives inverse Hilbert matric explicitly).

Crossrefs

Cf. A000290, A000515, A144631.

Programs

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

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 *)

Extensions

More terms from R. J. Mathar and Klaus Brockhaus, Jan 21 2009