A011195 - OEIS

%I #26 Oct 21 2022 21:11:31

%S 0,4,35,140,390,880,1729,3080,5100,7980,11935,17204,24050,32760,43645,

%T 57040,73304,92820,115995,143260,175070,211904,254265,302680,357700,

%U 419900,489879,568260,655690,752840,860405,979104,1109680,1252900,1409555,1580460,1766454

%N a(n) = n*(n+1)*(2*n+1)*(3*n+1)/6.

%H Ivan Panchenko, <a href="/A011195/b011195.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: x*(4 +15*x +5*x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

%F E.g.f.: x*(24 + 81*x + 47*x^2 + 6*x^3)*exp(x)/6. - _G. C. Greubel_, Mar 03 2020

%F From _Amiram Eldar_, Mar 10 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 36 - 9*sqrt(3)*Pi/2 + 48*log(2) - 81*log(3)/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (9*sqrt(3)-12)*Pi - 36*(1-log(2)). (End)

%p seq( n*mul(j*n+1, j=1..3)/6, n=0..40); # _G. C. Greubel_, Mar 03 2020

%t Table[n(n+1)(2n+1)(3n+1)/6,{n,0,40}]  (* _Harvey P. Dale_, Feb 24 2011 *)

%o (PARI) vector(41, n, my(m=n-1); m*prod(j=1,3, j*m+1)/6) \\ _G. C. Greubel_, Mar 03 2020

%o (Magma) [n*(&*[j*n+1:j in [1..3]])/6: n in [0..40]]; // _G. C. Greubel_, Mar 03 2020

%o (Sage) [n*product(j*n+1 for j in (1..3))/6 for n in (0..40)] # _G. C. Greubel_, Mar 03 2020

%o (GAP) List([0..40], n-> n*(n+1)*(2*n+1)*(3*n+1)/6 ); # _G. C. Greubel_, Mar 03 2020

%Y Cf. A094323.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_