A075913 - OEIS

%I #17 Dec 12 2015 13:13:25

%S 1,75,3500,131250,4344375,132890625,3855156250,107765625000,

%T 2933008203125,78271552734375,2058270703125000,53524929199218750,

%U 1380066321044921875,35349237725830078125,900813505310058593750,22863955398559570312500,578500758117828369140625

%N Fifth column of triangle A075500.

%C The e.g.f. given below is Sum_{m=0..4}(A075513(5,m)*exp(5*(m+1)*x))/4!.

%H Colin Barker, <a href="/A075913/b075913.txt">Table of n, a(n) for n = 0..714</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (75,-2125,28125,-171250,375000).

%F a(n) = A075500(n+5, 5) = (5^n)*S2(n+5, 5) with S2(n, m) = A008277(n, m) (Stirling2).

%F a(n) = Sum_{m=0..4}(A075513(5, m)*((m+1)*5)^n)/4!.

%F G.f.: 1/Product_{k=1..5}(1-5*k*x).

%F E.g.f.: (d^5/dx^5)((((exp(5*x)-1)/5)^5)/5!) = (exp(5*x) - 64*exp(10*x) + 486*exp(15*x) - 1024*exp(20*x) + 625*exp(25*x))/4!.

%F G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)). - _Colin Barker_, Dec 12 2015

%t Table[5^n*(1 - 2^(n+6) + 2*3^(n+5) - 4^(n+5) + 5^(n+4))/24, {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 12 2015 *)

%o (PARI) Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + O(x^30)) \\ _Colin Barker_, Dec 12 2015

%Y Cf. A000351, A016164, A075911, A075912, A075914.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Oct 02 2002