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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.

%I A144630 #15 Sep 08 2022 08:45:38
%S A144630 1,12,4,180,12,9,2800,880,40,16,44100,46900,4480,40,25,698544,1615824,
%T A144630 411264,13104,84,36,11099088,45094896,23653476,2268756,36036,84,49,
%U A144630 176679360,1115345088,1017615456,207193536,9660816,79776,144,64
%N 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.
%C A144630 The initial entries in each row form A000515. The second entries give A144631. The final entries are the squares (A000290).
%C A144630 Row sums are A144632. The penultimate entries in each row appear to be 4*A014105. - _N. J. A. Sloane_, Jan 20 2009
%H A144630 Klaus Brockhaus, <a href="/A144630/b144630.txt">Table of n, a(n) for n=1..1830 (rows 1 - 60)</a>
%H A144630 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hilbert_matrix">Hilbert matrix</a> (gives inverse Hilbert matric explicitly).
%e A144630 The first three inverse Hilbert matrices are:
%e A144630 --------------
%e A144630 [ 1 ]
%e A144630 --------------
%e A144630 [4 -6 ]
%e A144630 [-6 12]
%e A144630 --------------
%e A144630 [ 9 -36 30 ]
%e A144630 [-36 192 -180]
%e A144630 [30 -180 180]
%e A144630 --------------
%e A144630 Triangle begins:
%e A144630 1,
%e A144630 12, 4,
%e A144630 180, 12, 9,
%e A144630 2800, 880, 40, 16,
%e A144630 44100, 46900, 4480, 40, 25,
%e A144630 698544, 1615824, 411264, 13104, 84, 36
%p A144630 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
%t A144630 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 *)
%o A144630 (MATLAB) invhilb(1), invhilb(2), invhilb(3), etc.
%o A144630 (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
%Y A144630 Cf. A000290, A000515, A144631.
%K A144630 nonn,tabl
%O A144630 1,2
%A A144630 Daniel McLaury and Ben Golub, Dec 23 2008
%E A144630 More terms from _R. J. Mathar_ and _Klaus Brockhaus_, Jan 21 2009