OFFSET
0,7
COMMENTS
Essentially Stirling Numbers of the Second Kind, with an offset index, and multiplied by 720.
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)].
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).
FORMULA
a(n) = 720 * S(n+1,7), n>=0.
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
E.g.f.: Sum_{k=1..7} (-1)^(7-k)*binomial(7-1,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}, Table[row[n], {n, 6, 23}] [[All, 7]]] (* Jean-François Alcover, Dec 16 2014 *)
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 *)
Table[6!*StirlingS2[n + 1, 7], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
PROG
(PARI) a(n)=7^(n)-6*6^(n)+15*5^(n)-20*4^(n)+15*3^(n)-6*2^(n)+1
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard V. Scholtz, III, Sep 07 2013
EXTENSIONS
Offset corrected by Jean-François Alcover, Dec 16 2014
a(20) corrected by Jean-François Alcover, Dec 16 2014
Formula adapted for new offset by Vaclav Kotesovec, Dec 16 2014
STATUS
approved