A249076 - OEIS

%I #29 Sep 08 2022 08:46:10

%S 0,64,46656,2985984,64000000,729000000,5489031744,30840979456,

%T 139314069504,531441000000,1771561000000,5289852801024,14412774445056,

%U 36343632130624,85766121000000,191102976000000,404961208827904,820972403643456,1600135042849344,3010936384000000,5489031744000000

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

%H Andrew Howroyd, <a href="/A249076/b249076.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

%F a(n) = A002378(n)^6.

%F a(n) = 64*A059978(n) for n>0.

%F G.f.: 64*x*(x^10 + 716*x^9 + 37257*x^8 + 450048*x^7 + 1822014*x^6 + 2864328*x^5 + 1822014*x^4 + 450048*x^3 + 37257*x^2 + 716*x + 1)/(1 - x)^13. [corrected by _Georg Fischer_, May 10 2019]

%F Sum_{n>=1} 1/a(n) = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945. - _Vaclav Kotesovec_, Sep 25 2019

%p [ seq(n^6*(n+1)^6, n = 0..100) ];

%t Table[(n (n + 1))^6, {n, 0, 70}] (* or *)

%t CoefficientList[Series[64*x*(x^10 + 716 x^9 + 37257 x^8 + 450048 x^7 + 1822014 x^6 + 2864328 x^5 + 1822014 x^4 + 450048 x^3 + 37257 x^2 + 716 x + 1)/(1 - x)^13, {x, 0, 30}], x]

%o (Magma) [(n*(n+1))^6: n in [0..30]];

%o (PARI) a(n)=(n*(n+1))^6 \\ _Charles R Greathouse IV_, Oct 21 2014

%Y Cf. A059978; A002378: n*(n+1); A035282: n^2 *(n+1)^2; A060459: n^3 *(n+1)^3; A248619: n^4 *(n+1)^4;

%K nonn,easy

%O 0,2

%A _Jiwoo Lee_, Oct 20 2014

%E Incorrect term corrected by _Colin Barker_, Oct 21 2014

%E Terms a(21) and beyond corrected by _Andrew Howroyd_, Feb 22 2018