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A119537 Determinant of n X n matrices of first n^2 denumerants (A000115).

%I A119537 #18 Sep 13 2024 06:52:31
%S A119537 1,0,-3,-3,0,-54,343,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A119537 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A119537 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A119537 Determinant of n X n matrices of first n^2 denumerants (A000115).
%C A119537 Conjecture: a(n>7)=0. - _Robert G. Wilson v_, Jun 07 2006
%F A119537 a(n) = determinant[A000115(k) from k=1 to n^2].
%e A119537 a(6) = -54 = -2 * 3^3. a(7) = 343 = 7^3.
%e A119537 a(8) = 0 because of the singular matrix 0 =
%e A119537 |..1...1...2...2...3...4...5...6|
%e A119537 |..7...8..10..11..13..14..16..18|
%e A119537 |.20..22..24..26..29..31..34..36|
%e A119537 |.39..42..45..48..51..54..58..61|
%e A119537 |.65..68..72..76..80..84..88..92|
%e A119537 |.97.101.106.110.115.120.125.130|
%e A119537 |135.140.146.151.157.162.168.174|
%e A119537 |180.186.192.198.205.211.218.224|.
%t A119537 clst = CoefficientList[ Series[1/((1 - x)(1 - x^2)(1 - x^5)), {x, 0, 105^2 - 1}], x];
%t A119537 f[n_] := Det[ Partition[ Take[clst, n^2], n]];
%t A119537 Array[f,100] (* _Robert G. Wilson v_, Jun 07 2006 *)
%Y A119537 Cf. A000115, A119493.
%K A119537 easy,sign
%O A119537 1,3
%A A119537 _Jonathan Vos Post_, May 28 2006
%E A119537 More terms from _Robert G. Wilson v_, Jun 07 2006