|
%I #15 Dec 12 2015 13:16:11
%S 1,50,1625,43750,1063125,24281250,532890625,11386718750,238867578125,
%T 4946347656250,101481884765625,2068161621093750,41943091064453125,
%U 847579699707031250,17082562164306640625,343617765808105468750,6901873153839111328125
%N Fourth column of triangle A075500.
%C The e.g.f. given below is (Sum_{m=0..3} A075513(4,m)*exp(5*(m+1)*x))/3!.
%H Colin Barker, <a href="/A075912/b075912.txt">Table of n, a(n) for n = 0..767</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (50,-875,6250,-15000).
%F a(n) = A075500(n+4, 4) = (5^n)*S2(n+4, 4) with S2(n, m) = A008277(n, m) (Stirling2).
%F a(n) = (Sum_{m=0..3}A075513(4, m)*((m+1)*5)^n, m=0..3)/3!.
%F G.f.: 1/Product_{k=1..4}(1-5*k*x).
%F E.g.f.: (d^4/dx^4)((((exp(5*x)-1)/5)^4)/4!) = (-exp(5*x) + 24*exp(10*x) - 81*exp(15*x) + 64*exp(20*x))/3!.
%F a(n) = 50*a(n-1) - 875*a(n-2) + 6250*a(n-3) - 15000*a(n-4) for n>3. - _Colin Barker_, Dec 11 2015
%t Table[5^n*(-1 + 3*2^(3+n) + 2^(6+2*n) - 3^(4+n))/6, {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 12 2015 *)
%o (PARI) Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)) + O(x^30)) \\ _Colin Barker_, Dec 11 2015
%Y Cf. A075911, A075913.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 02 2002
|
