

A124261


Sequence terms are generated by solving the n x n linear algebra problem [H]x = b, where b is the unit vector. Only xn, the last unknown is used.


1



1, 1, 2, 3, 12, 20, 300, 525, 1960, 49392, 1481760, 5821200, 164656800, 336370320, 3741185448
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OFFSET

1,3


COMMENTS

These matrices are notorious for being illconditioned, they are solved using rational arithmetic. To reproduce these numbers, or larger ones, one can edit the program above with a text editor and then drag it into the Maxima window. The last term required approx. 10 minutes to generate on an Intel dualcore 6600 clocked at 2.4 GHz with 2 Gig of RAM. To change the order of the Hilbert matrix, just change the order variable at the top of the file. It is currently 4, which runs quickly. So a point of curiosity really. The numbers become extremely expensive to find, growing at least as the cube of n times a large constant. So to me they are a kind of symbolic gold. The 100th number for example, is probably not knowable at the current time, but that is speculation on my part. Perhaps you will observe some functional relationship that allows their simple generation thus "Cracking the HilbertWarren Sequence Code". For example 12 = 4 * 4  4, 20 = 5 * 5  5, 300 = 20 * 20  100, 525 = 25 * 25  100 and so on.


LINKS

Table of n, a(n) for n=1..15.
William Schelter, DOE Maxima  A Symbolic Algebra Program
Gilbert Strang, Linear Algebra and Its Applications
L. V. Warren, Cracking the HilbertWarren Code


FORMULA

Solve the n x n linear algebra problem [H]x = b, where b is the unit vector and n is the order. Only xn, the last unknown is used.


EXAMPLE

Triangularization of [H] is performed, followed by inversion or back substitution. The terms are the denominators of the last solution xn to the Hilbert matrices of order 1, 2, ... 15, respectively. The numerator is unity in all cases.
For order 3 problem the solutions are: ([1/6],[ 1/3],[1/2]), thus a() = 2, since we just take the denominator.


PROG

load("eigen"); order : 4; X : columnvector(makelist(concat(x, i), i, 1, order)); h[i, j] := 1/(i + j 1); Unity[i, j] := 1; A : genmatrix(h, order, order); A . X; B : genmatrix(Unity, 1, order); A . X = B; Ap : triangularize(A); Ap . X = B; App : invert(Ap); App . B;


CROSSREFS

Cf. A124262.
Sequence in context: A205825 A076000 A096632 * A018883 A066730 A077755
Adjacent sequences: A124258 A124259 A124260 * A124262 A124263 A124264


KEYWORD

eigen,frac,hard,nonn


AUTHOR

L. Van Warren (van(AT)wdv.com), Oct 23 2006


EXTENSIONS

The author said that this version of the submission was unsatisfactory and set me a better version in email, which may replace this.  N. J. A. Sloane, Oct 24 2006


STATUS

approved



