A071244 - OEIS

%I #23 Aug 07 2024 01:14:09

%S 0,0,2,11,36,90,190,357,616,996,1530,2255,3212,4446,6006,7945,10320,

%T 13192,16626,20691,25460,31010,37422,44781,53176,62700,73450,85527,

%U 99036,114086,130790,149265,169632,192016,216546,243355,272580,304362,338846,376181

%N a(n) = n*(n-1)*(n^2 + 2)/6.

%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

%H Vincenzo Librandi, <a href="/A071244/b071244.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4, a(0)=0, a(1)=0, a(2)=2, a(3)=11, a(4)=36. - _Yosu Yurramendi_, Sep 03 2013

%F From _G. C. Greubel_, Aug 06 2024: (Start)

%F G.f.: x^2*(2 + x + x^2)/(1 - x)^5.

%F E.g.f.: (1/6)*x^2*(6 + 5*x + x^2)*exp(x). (End)

%t Table[n(n-1)(n^2+2)/6,{n,0,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,2,11,36},50] (* _Harvey P. Dale_, Nov 27 2022 *)

%o (Magma) [n*(n-1)*(n^2+2)/6: n in [0..40]]; // _Vincenzo Librandi_, Jun 14 2011

%o (PARI) a(n)=n*(n-1)*(n^2+2)/6; \\ _Joerg Arndt_, Sep 04 2013

%o (SageMath)

%o def A071244(n): return binomial(n,2)*(n^2+2)//3

%o [A071244(n) for n in range(41)] # _G. C. Greubel_, Aug 06 2024

%Y Cf. A071239.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 12 2002