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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.
2
3, 20, 420, 18144, 1330560, 148262400, 23351328000, 4940103168000, 1351612226764800, 464463110651904000, 195848611658219520000, 99430833611096064000000, 59828953024276660224000000, 42103628541617628354969600000, 34261827725741345073856512000000, 31923961833867229762934538240000000
OFFSET
2,1
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)
FORMULA
a(n) = (2n - 1)!/(n*(n - 1)).
Sum_{n=2..oo} 1/a(n) = (1 + e^2)/(8*e) = 0.38577015870381094461947640518926542... . - Stefano Spezia, Jul 27 2024
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}]
PROG
(PARI) a(n) = (2*n - 1)!/(n*(n - 1)); \\ Michel Marcus, Oct 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Apr 02 2008
EXTENSIONS
More terms from Michel Marcus, Oct 28 2017
STATUS
approved