A099764 a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

0, 36, 576, 3600, 14400, 44100, 112896, 254016, 518400, 980100, 1742400, 2944656, 4769856, 7452900, 11289600, 16646400, 23970816, 33802596, 46785600, 63680400, 85377600, 112911876, 147476736, 190440000, 243360000, 308002500, 386358336

Offset: 0

Author

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

Keywords

• easy
• nonn

Examples

			a(0) = 1^3 - 1^3 = 0;
a(1) = (1+3)^3 - (1^3+3^3) = 64 - 28 = 36;
a(2) = (1+3+5)^3 - (1^3+3^3+5^3) = 729 - 153 = 576;
a(3) = (1+3+5+7)^3 - (1^3+3^3+5^3+7^3) = 4096 - 496 = 3600;
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
		

References

• Jolley, Summation of Series, Dover (1961).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A001014, A002593.

Programs

GAP

List([1..30], n-> (n*(n^2-1))^2); # G. C. Greubel, Sep 03 2019

Magma

[n^2*(n+1)^2*(n+2)^2: n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

Maple

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

Mathematica

Table[n^2*(n+1)^2*(n+2)^2, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
Times@@@Partition[Range[0,30]^2,3,1] (* Harvey P. Dale, Sep 02 2016 *)

PARI

vector(30, n, (n*(n^2-1))^2) \\ G. C. Greubel, Sep 03 2019

Sage

[(n*(n^2-1))^2 for n in (1..30)] # G. C. Greubel, Sep 03 2019

Formula

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

G.f.: 36*x*(1+x)*(1 +8*x +x^2)/(1-x)^7 . - R. J. Mathar, Oct 03 2011

a(n) = (Sum_{k=0..n} (2*k+1))^3 - Sum_{k=0..n} (2*k+1)^3. - Philippe Deléham, Mar 10 2014

a(n) = A001014(n+1) - A002593(n+1). - Philippe Deléham, Mar 10 2014

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

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