A099764 - OEIS

%I #41 Sep 08 2022 08:45:15

%S 0,36,576,3600,14400,44100,112896,254016,518400,980100,1742400,

%T 2944656,4769856,7452900,11289600,16646400,23970816,33802596,46785600,

%U 63680400,85377600,112911876,147476736,190440000,243360000,308002500,386358336

%N a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

%D Jolley, Summation of Series, Dover (1961).

%H Vincenzo Librandi, <a href="/A099764/b099764.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F Sum_{n>=1} 1/a(n) = Pi^2/4-39/16 =  0.029901100272... [Jolley eq 241]

%F G.f.: 36*x*(1+x)*(1 +8*x +x^2)/(1-x)^7 . - _R. J. Mathar_, Oct 03 2011

%F a(n) = (Sum_{k=0..n} (2*k+1))^3 - Sum_{k=0..n} (2*k+1)^3. - _Philippe Deléham_, Mar 10 2014

%F a(n) = A001014(n+1) - A002593(n+1). - _Philippe Deléham_, Mar 10 2014

%F E.g.f.: exp(x)*x*(36+252*x+330*x^2+138*x^3+21*x^4+x^5). - _Stefano Spezia_, Sep 04 2019

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 - 7/16. - _Amiram Eldar_, Jul 02 2020

%e a(0) = 1^3 - 1^3 = 0;

%e a(1) = (1+3)^3 - (1^3+3^3) = 64 - 28 = 36;

%e a(2) = (1+3+5)^3 - (1^3+3^3+5^3) = 729 - 153 = 576;

%e a(3) = (1+3+5+7)^3 - (1^3+3^3+5^3+7^3) = 4096 - 496 = 3600;

%e a(4) = (1+3+5+7+9)^3 - (1^3+3^3+5^3+7^3+9^3) = 15625 - 1225 = 14400; etc. - _Philippe Deléham_, Mar 10 2014

%p A099764:=n->n^2*(n+1)^2*(n+2)^2; seq(A099764(n), n=0..30); # _Wesley Ivan Hurt_, Mar 09 2014

%t Table[n^2*(n+1)^2*(n+2)^2, {n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2011 *)

%t Times@@@Partition[Range[0,30]^2,3,1] (* _Harvey P. Dale_, Sep 02 2016 *)

%o (Magma) [n^2*(n+1)^2*(n+2)^2: n in [0..30]]; // _Vincenzo Librandi_, Oct 04 2011

%o (PARI) vector(30, n, (n*(n^2-1))^2) \\ _G. C. Greubel_, Sep 03 2019

%o (Sage) [(n*(n^2-1))^2 for n in (1..30)] # _G. C. Greubel_, Sep 03 2019

%o (GAP) List([1..30], n-> (n*(n^2-1))^2); # _G. C. Greubel_, Sep 03 2019

%Y Cf. A001014, A002593.

%K easy,nonn

%O 0,2

%A Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004