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%I #50 Sep 08 2022 08:46:05
%S 0,0,0,0,0,0,0,5040,181440,3780000,59875200,801496080,9574044480,
%T 105398092800,1091804313600,10794490827120,102896614941120,
%U 952741767650400,8617145057539200,76461500619902160,667855517349303360,5757691363157764800,49099453300298016000,414884142077935345200
%N a(n) = 8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1.
%C Calculates the eighth 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 Seiichi Manyama, <a href="/A228910/b228910.txt">Table of n, a(n) for n = 0..1107</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).
%F a(n) = 5040 * S2(n+1,8), n>=0.
%F G.f.: 5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - _Colin Barker_, Sep 20 2013
%F E.g.f.: Sum_{k=1..8} (-1)^(8-k)*binomial(8-1,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, 0}, Table[ -row[n], {n, 7, 23}] [[All, 8]]] (* _Jean-François Alcover_, Dec 16 2014 *)
%t Table[7!*StirlingS2[n + 1, 8], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%t Table[8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1, {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)
%t CoefficientList[Series[5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 16 2014 after _Colin Barker_ *)
%o (PARI) a(n)=8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1.
%o (PARI) for(n=0,30, print1(5040*stirling(n+1,8,2), ", ")) \\ _G. C. Greubel_, Nov 19 2017
%o (Magma) [8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1: n in [0..30]]; // _G. C. Greubel_, Nov 19 2017
%Y Cf. A008277, A049434.
%Y The eighth column of results of A163626.
%Y Cf. A000225, A028243, A028244, A028245, A032180, A228909.
%K nonn,easy
%O 0,8
%A _Richard V. Scholtz, III_, Sep 07 2013
%E Offset corrected by _Vaclav Kotesovec_, Dec 16 2014
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