cp's OEIS Frontend

A272871 Imaginary part of (n + i)^4.

%I A272871 #31 Oct 30 2019 01:32:14
%S A272871 0,0,24,96,240,480,840,1344,2016,2880,3960,5280,6864,8736,10920,13440,
%T A272871 16320,19584,23256,27360,31920,36960,42504,48576,55200,62400,70200,
%U A272871 78624,87696,97440,107880,119040,130944,143616,157080,171360,186480,202464,219336
%N A272871 Imaginary part of (n + i)^4.
%H A272871 Colin Barker, <a href="/A272871/b272871.txt">Table of n, a(n) for n = 0..1000</a>
%H A272871 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A272871 a(n) = 4*A007531(n+1).
%F A272871 a(n) = 4*(n-1)*n*(n+1).
%F A272871 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
%F A272871 G.f.: 24*x^2 / (1-x)^4.
%F A272871 a(n) = b(n+1)*b(n-1)-b(n)*b(n-2), where b(n) is A002378(n). - _Anton Zakharov_, Aug 15 2016
%F A272871 From _Ilya Gutkovskiy_, Aug 15 2016: (Start)
%F A272871 E.g.f.: 4*x^2*(3 + x)*exp(x).
%F A272871 a(n) = 24*binomial(n+1,3).
%F A272871 a(n) = Sum_{k=0..n} A064200(k). (End)
%e A272871 a(5) = 480 because (5 + i)^4 = 476 + 480*i.
%t A272871 Table[Im[(n + I)^4], {n, 0, 38}] (* or *)
%t A272871 Table[4 (n - 1) n (n + 1), {n, 0, 38}] (* or *)
%t A272871 CoefficientList[Series[24 x^2/(1 - x)^4, {x, 0, 38}], x] (* _Michael De Vlieger_, May 08 2016 *)
%o A272871 (PARI) a(n) = 4*(n-1)*n*(n+1)
%o A272871 (PARI) vector(50, n, n--; imag((n+I)^4))
%o A272871 (PARI) concat(vector(2), Vec(24*x^2/(1-x)^4 + O(x^50)))
%Y A272871 Cf. A002378, A007531, A064200.
%Y A272871 Cf. A272870.
%K A272871 nonn,easy
%O A272871 0,3
%A A272871 _Colin Barker_, May 08 2016