|
%I #19 Nov 26 2022 12:53:20
%S 0,0,3,15,48,121,261,504,896,1494,2367,3597,5280,7527,10465,14238,
%T 19008,24956,32283,41211,51984,64869,80157,98164,119232,143730,172055,
%U 204633,241920,284403,332601,387066,448384,517176,594099,679847,775152,880785,997557
%N a(n) = n*(n-1)*(n^3 + 21*n^2 - 4*n + 96)/120.
%C Arises in studying the Goldbach conjecture.
%H Matthew House, <a href="/A124161/b124161.txt">Table of n, a(n) for n = 0..10000</a>
%H P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-23/1/290.extract">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [See p. 301]
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: x^2*(3-3*x+3*x^2-2*x^3)/(1-x)^6. - _Matthew House_, Jan 16 2017
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - _Colin Barker_, Jan 16 2017
%t Table[n(n-1)(n^3+21n^2-4n+96)/120,{n,0,50}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,3,15,48,121},50] (* _Harvey P. Dale_, Nov 26 2022 *)
%o (PARI) concat(vector(2), Vec(x^2*(3-3*x+3*x^2-2*x^3) / (1-x)^6 + O(x^40))) \\ _Colin Barker_, Jan 16 2017
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 03 2006
|
