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

3, 20, 420, 18144, 1330560, 148262400, 23351328000, 4940103168000, 1351612226764800, 464463110651904000, 195848611658219520000, 99430833611096064000000, 59828953024276660224000000, 42103628541617628354969600000, 34261827725741345073856512000000, 31923961833867229762934538240000000

Offset: 2

Author

Artur Jasinski, Apr 02 2008

Keywords

• nonn

Comments

From Anthony Hernandez, Oct 24 2017: (Start)

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:

(6,5) = 7 6 5 4 3 1

5 4 3 2 1

and there are a(6-1) = a(5) = 1330560 permutations with shape (6,5). (End)

Crossrefs

Cf. A002378, A009445, A007878, A138896, A138898, A000108.

Programs

Maple

A138897:=n->(2*n - 1)!/(n*(n - 1)): seq(A138897(n), n=2..20); # Wesley Ivan Hurt, Nov 25 2017

Mathematica

Table[(2 n - 1)!/(n (n - 1)), {n, 2, 20}]

PARI

a(n) = (2*n - 1)!/(n*(n - 1)); \\ Michel Marcus, Oct 28 2017

Formula

a(n) = (2n - 1)!/(n*(n - 1)).

Sum_{n>=2} 1/a(n) = (1 + e^2)/(8*e) = 0.38577015870381094461947640518926542... . - Stefano Spezia, Jul 27 2024

Sum_{n>=2} (-1)^n/a(n) = (2*sin(1) - cos(1))/4. - Amiram Eldar, Aug 19 2025

Extensions

More terms from Michel Marcus, Oct 28 2017