A138896 -id:A138896 - cp's OEIS frontend

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

Artur Jasinski, Apr 02 2008

nonn

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)

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

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

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

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

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

More terms from Michel Marcus, Oct 28 2017

A138898 Ratio of (2*n-1)! to number of zeros in lower part of Sylvester matrix for polynomial of degree n with all nonzero coefficients.

Original entry on oeis.org

60, 840, 30240, 1995840, 207567360, 31135104000, 6351561216000, 1689515283456000, 567677135241216000, 235018333989863424000, 117509166994931712000000, 69800445194989436928000000, 48581109855712648101888000000, 39156374543704394370121728000000, 36180490078382860397992476672000000, 37989514582302003417892100505600000000, 44979585265445572046784246998630400000000, 59642930061980828534035911520183910400000000

Offset: 3

Artur Jasinski, Apr 02 2008

Cf. A002378, A009445, A007878, A138896.

Table[(2 n - 1)!/((n - 1)(n - 2)), {n, 3, 20}] (* R. J. Mathar, Apr 30 2008 *)

a(n) = (2*n-1)!/((n-1)*(n-2)). - R. J. Mathar, Apr 30 2008
Sum_{n=3..oo} 1/a(n) = (3*cosh(1) + 10*sinh(1) - 6*e)/4 = 0.0178907175323686230239526350278045532... . - Stefano Spezia, Jul 27 2024
Equivalently, Sum_{n=3..oo} 1/a(n) = (e^2 - 7)/(8*e). - Vaclav Kotesovec, Aug 19 2025
Sum_{n>=3} (-1)^(n+1)/a(n) = (2*sin(1) - 3*cos(1))/4. - Amiram Eldar, Aug 19 2025

Edited and corrected by R. J. Mathar, Apr 30 2008

cp's OEIS Frontend

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

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