A071245 - OEIS

%I #57 Aug 07 2024 01:21:00

%S 0,0,3,19,66,170,365,693,1204,1956,3015,4455,6358,8814,11921,15785,

%T 20520,26248,33099,41211,50730,61810,74613,89309,106076,125100,146575,

%U 170703,197694,227766,261145,298065,338768,383504,432531,486115,544530,608058

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

%C The first differences are given in A277228. - _J. M. Bergot_, Sep 14 2016

%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="/A071245/b071245.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) for n > 4; a(0)=0, a(1)=0, a(2)=3, a(3)=19, a(4)=66. - _Yosu Yurramendi_, Sep 03 2013

%F G.f.: x^2*(3 + 4*x + x^2)/(1-x)^5. - _Michael De Vlieger_, Sep 14 2016

%F E.g.f.: (1/6)*x^2*(9 + 10*x + 2*x^2)*exp(x). - _G. C. Greubel_, Sep 23 2016

%t Table[n (n - 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)

%t CoefficientList[Series[(-3 x^2 - 4 x^3 - x^4)/(-1 + x)^5, {x, 0, 37}], x] (* _Michael De Vlieger_, Sep 14 2016 *)

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

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

%o (SageMath)

%o def A071245(n): return binomial(n,2)*(2*n^2+1)//3

%o [A071245(n) for n in range(41)] # _G. C. Greubel_, Aug 07 2024

%Y Cf. A071238, A071244, A277228 (first differences).

%K nonn,easy

%O 0,3

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