OFFSET
0,9
COMMENTS
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)].
Essentially 40320 * A049447. - Joerg Arndt, Sep 24 2016
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1047
Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).
FORMULA
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
E.g.f.: Sum_{k=1..9} (-1)^(9-k)*binomial(9-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017
MATHEMATICA
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 *)
Table[8!*StirlingS2[n + 1, 9], {n, 0, 22}] (* Vaclav Kotesovec, Dec 16 2014 *)
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 *)
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 *)
lst={}; Do[f= 40320 StirlingS2[n, 9]; AppendTo[lst, f], {n, 1, 5!}]; lst (* Vincenzo Librandi, Oct 11 2017 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard V. Scholtz, III, Sep 07 2013
EXTENSIONS
Offset corrected by Vaclav Kotesovec, Dec 16 2014
STATUS
approved