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
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).
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
Comments