%I #44 Sep 08 2022 08:45:16
%S 0,2,48,324,1280,3750,9072,19208,36864,65610,110000,175692,269568,
%T 399854,576240,810000,1114112,1503378,1994544,2606420,3360000,4278582,
%U 5387888,6716184,8294400,10156250,12338352,14880348,17825024,21218430,25110000,29552672
%N a(n) = (n+1)*n^4.
%C For n>=4, a(n-1) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that for fixed, different x_1, x_2, x_3, x_4 in {1,2,3,4,5} and fixed y_1, y_2, y_3, y_ 4 in {1,2,...n} we have f(x_i)<>y_i, (i=1,2,3,4). - _Milan Janjic_, May 13 2007
%C Pierce expansion of the constant 1 - Sum_{k >= 1} (-1)^(k+1)*k^4/k!^5 = 0.48961 54584 28443 62043 ... = 1/2 - 1/(2*48) + 1/(2*48*324) - .... - _Peter Bala_, Feb 01 2015
%H Vincenzo Librandi, <a href="/A101362/b101362.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Katherine Kanim, <a href="http://www.jstor.org/stable/3219288">Proof without Words: The Sum of Cubes: An Extension of Archimedes' Sum of Squares</a>, Mathematics Magazine, Vol. 77, No. 4 (2004), pp. 298-299.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) + 6*Sum_{i=1..n} i^3 + 4*Sum_{i=1..n} i^2 + Sum_{i=1..n} i = 5*Sum_{i=1..n} i^4.
%F G.f.: 2*x*(8*x^3+33*x^2+18*x+1) / (x-1)^6. - _Colin Barker_, May 06 2013
%F Sum_{n>=1} 1/a(n) = 0.5252003... = Pi^2/6+Pi^4/90-Zeta(3)-1. - _R. J. Mathar_, Oct 18 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - 2*log(2) + Pi^2/12 - 3*zeta(3)/4 + 7*Pi^4/720. - _Amiram Eldar_, Nov 05 2020
%e a(5) = (5+1)*5^4 = 3750 = 2 * 3 * 5^4, the sum of the divisors of which is 30008.
%e a(7) = 8*7^4 = 19208 = 2^3 * 7^4 = 98^2 + 98^2.
%e a(8) = 9*8^4 = 36864 = 2^12*3^2 = 192^2.
%e a(9) = 10*9^4 = 65610 = 2*3^8*5 = 243^2 + 81^2.
%e a(10) = 11*10^4 = 110000 = 2^4*5^4*11 = 300^2 + 100^2 + 100^2.
%p a:= n-> (n+1)*n^4: seq(a(n), n=0..35);
%t Table[(n + 1)*n^4, {n, 0, 30}]
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,48,324,1280,3750},40] (* _Harvey P. Dale_, Jun 10 2019 *)
%o (Magma) [n^4+n^5: n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2016
%Y Cf. A019583.
%K nonn,easy
%O 0,2
%A _Jonathan Vos Post_, Dec 25 2004
%E Corrected and extended by _Ray Chandler_, Dec 26 2004