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%I A228911 #44 Sep 08 2022 08:46:05
%S A228911 0,0,0,0,0,0,0,0,40320,1814400,46569600,898128000,14495120640,
%T A228911 207048441600,2706620716800,33094020960000,384202115256960,
%U A228911 4280991956841600,46150861752777600,484294916235312000,4970346251077025280,50075960398487654400,496745174491651008000
%N A228911 a(n) = 9^n - 8*8^n + 28*7^n - 56*6^n + 70*5^n - 56*4^n + 28*3^n - 8*2^n + 1.
%C A228911 Calculates the ninth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].
%C A228911 Essentially 40320 * A049447. - _Joerg Arndt_, Sep 24 2016
%H A228911 Seiichi Manyama, <a href="/A228911/b228911.txt">Table of n, a(n) for n = 0..1047</a>
%H A228911 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).
%F A228911 G.f.: -40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - _Colin Barker_, Sep 20 2013
%F A228911 E.g.f.: Sum_{k=1..9} (-1)^(9-k)*binomial(9-1,k-1)*exp(k*x). - _Wolfdieter Lang_, May 03 2017
%t A228911 Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0, 0, 0}, Table[row[n], {n, 8, 22}] [[All, 9]]] (* _Jean-François Alcover_, Dec 16 2014 *)
%t A228911 Table[8!*StirlingS2[n + 1, 9], {n, 0, 22}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%t A228911 Table[9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1, {n, 0, 22}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%t A228911 CoefficientList[Series[-40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 16 2014 after _Colin Barker_ *)
%t A228911 lst={}; Do[f= 40320 StirlingS2[n, 9];  AppendTo[lst, f], {n, 1, 5!}]; lst (* _Vincenzo Librandi_, Oct 11 2017 *)
%o A228911 (PARI) a(n)=9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1
%o A228911 (Magma) [9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1: n in [0..32]]; // _Vincenzo Librandi_, Oct 11 2017
%Y A228911 The ninth column of results of A163626.
%Y A228911 Cf. A228910 (also for more crossrefs).
%K A228911 nonn,easy
%O A228911 0,9
%A A228911 _Richard V. Scholtz, III_, Sep 07 2013
%E A228911 Offset corrected by _Vaclav Kotesovec_, Dec 16 2014

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