A248720 - OEIS

%I #35 Jan 20 2024 14:40:29

%S 0,32,7776,248832,3200000,24300000,130691232,550731776,1934917632,

%T 5904900000,16105100000,40074642432,92389579776,199690286432,

%U 408410100000,796262400000,1488827973632,2682916351776,4678757435232,7923516800000,13069123200000

%N a(n) = (n*(n+1))^5.

%C This is the sequence (2^5)*A059860(n)= (2*binomial(n+1,2))^5, n >= 0. - _Wolfdieter Lang_, Nov 03 2014

%H Andrew Howroyd, <a href="/A248720/b248720.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

%F a(n) = A002378(n)^5.

%F a(n) = 32*A059860(n) for n>0.

%F G.f.: 32*x*(x^8 + 232*x^7 + 5158*x^6 + 27664*x^5 + 47290*x^4 + 27664*x^3 + 5158*x^2 + 232*x + 1) / (1 - x)^11 (from A059860).

%F Sum_{n>=1} 1/a(n) = 126 - 35*Pi^2/3 - Pi^4/9. - _Vaclav Kotesovec_, Sep 25 2019

%F a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - _Wesley Ivan Hurt_, Jan 20 2024

%p [ seq(n^5*(n+1)^5, n = 0..100) ];

%t Table[(n (n + 1))^5, {n, 0, 70}] (* or *) CoefficientList[Series[32 x (x^8 + 232 x^7 + 5158 x^6 + 27664 x^5 + 47290 x^4 + 27664 x^3 + 5158 x^2 + 232 x + 1)/(1 - x)^11, {x, 0, 30}], x]

%t LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,32,7776,248832,3200000,24300000,130691232,550731776,1934917632,5904900000,16105100000},20] (* _Harvey P. Dale_, Apr 23 2017 *)

%o (Magma) [(n*(n+1))^5: n in [0..30]];

%Y Cf. A059860, A002378 (n*(n+1)), A035287(n+1) ((n*(n+1))^2), A060459 ((n*(n+1))^3), A248619 ((n*(n+1))^4).

%K nonn,easy

%O 0,2

%A _Eugene Chong_, Oct 16 2014

%E Terms a(32) and beyond corrected by _Andrew Howroyd_, Feb 20 2018