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%I #45 Sep 08 2022 08:46:05
%S 0,0,0,0,0,0,720,20160,332640,4233600,46070640,451725120,4115105280,
%T 35517081600,294293759760,2362955474880,18509835445920,
%U 142172988048000,1074905737084080,8023358912869440,59263889194762560,433988913576556800,3155502239364459600,22807773973299268800
%N a(n) = 7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1.
%C Essentially Stirling Numbers of the Second Kind, with an offset index, and multiplied by 720.
%C Calculates the seventh 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)].
%H Colin Barker, <a href="/A228909/b228909.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-322,1960,-6769,13132,-13068,5040).
%F a(n) = 720 * S(n+1,7), n>=0.
%F G.f.: -720*x^6 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - _Colin Barker_, Dec 16 2014
%F E.g.f.: Sum_{k=1..7} (-1)^(7-k)*binomial(7-1,k-1)*exp(k*x). - _Wolfdieter Lang_, May 03 2017
%t 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}, Table[row[n], {n, 6, 23}] [[All, 7]]] (* _Jean-François Alcover_, Dec 16 2014 *)
%t Table[7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1, {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%t Table[6!*StirlingS2[n + 1, 7], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%o (PARI) a(n)=7^(n)-6*6^(n)+15*5^(n)-20*4^(n)+15*3^(n)-6*2^(n)+1
%o (PARI) concat([0,0,0,0,0,0], Vec(-720*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)) + O(x^100))) \\ _Colin Barker_, Dec 16 2014
%o (Magma) [7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1: n in [0..30]]; // _G. C. Greubel_, Nov 19 2017
%Y Cf. A000771, A008277.
%Y Represents the seventh column of results of A163626.
%K nonn,easy
%O 0,7
%A _Richard V. Scholtz, III_, Sep 07 2013
%E Offset corrected by _Jean-François Alcover_, Dec 16 2014
%E a(20) corrected by _Jean-François Alcover_, Dec 16 2014
%E Formula adapted for new offset by _Vaclav Kotesovec_, Dec 16 2014