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A138897 Ratio of (2n-1)! to number of zeros in upper part of Sylvester matrix of polynomial of degree n with all nonzero coefficients.

%I A138897 #48 Aug 19 2025 05:25:53
%S A138897 3,20,420,18144,1330560,148262400,23351328000,4940103168000,
%T A138897 1351612226764800,464463110651904000,195848611658219520000,
%U A138897 99430833611096064000000,59828953024276660224000000,42103628541617628354969600000,34261827725741345073856512000000,31923961833867229762934538240000000
%N A138897 Ratio of (2n-1)! to number of zeros in upper part of Sylvester matrix of polynomial of degree n with all nonzero coefficients.
%C A138897 From _Anthony Hernandez_, Oct 24 2017: (Start)
%C A138897 If (n,n-1) is the two-part partition of any odd integer greater than 1 then a(n-1) is the number of permutations of shape (n,n-1). For example, the two-part partition of 11 with shape (n,n-1) is (6,5). Pictorially we can draw this as a standard Young diagram with cells populated by hook lengths:
%C A138897           (6,5) = 7 6 5 4 3 1
%C A138897                   5 4 3 2 1
%C A138897   and there are a(6-1) = a(5) = 1330560 permutations with shape (6,5). (End)
%F A138897 a(n) = (2n - 1)!/(n*(n - 1)).
%F A138897 Sum_{n>=2} 1/a(n) = (1 + e^2)/(8*e) = 0.38577015870381094461947640518926542... . - _Stefano Spezia_, Jul 27 2024
%F A138897 Sum_{n>=2} (-1)^n/a(n) = (2*sin(1) - cos(1))/4. - _Amiram Eldar_, Aug 19 2025
%p A138897 A138897:=n->(2*n - 1)!/(n*(n - 1)): seq(A138897(n), n=2..20); # _Wesley Ivan Hurt_, Nov 25 2017
%t A138897 Table[(2 n - 1)!/(n (n - 1)), {n, 2, 20}]
%o A138897 (PARI) a(n) = (2*n - 1)!/(n*(n - 1)); \\ _Michel Marcus_, Oct 28 2017
%Y A138897 Cf. A002378, A009445, A007878, A138896, A138898, A000108.
%K A138897 nonn
%O A138897 2,1
%A A138897 _Artur Jasinski_, Apr 02 2008
%E A138897 More terms from _Michel Marcus_, Oct 28 2017