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.
60, 840, 30240, 1995840, 207567360, 31135104000, 6351561216000, 1689515283456000, 567677135241216000, 235018333989863424000, 117509166994931712000000, 69800445194989436928000000, 48581109855712648101888000000, 39156374543704394370121728000000, 36180490078382860397992476672000000, 37989514582302003417892100505600000000, 44979585265445572046784246998630400000000, 59642930061980828534035911520183910400000000
Offset: 3
Programs
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Mathematica
Table[(2 n - 1)!/((n - 1)(n - 2)), {n, 3, 20}] (* R. J. Mathar, Apr 30 2008 *)
Formula
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
Extensions
Edited and corrected by R. J. Mathar, Apr 30 2008