A228909 a(n) = 7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1.

0, 0, 0, 0, 0, 0, 720, 20160, 332640, 4233600, 46070640, 451725120, 4115105280, 35517081600, 294293759760, 2362955474880, 18509835445920, 142172988048000, 1074905737084080, 8023358912869440, 59263889194762560, 433988913576556800, 3155502239364459600, 22807773973299268800

Offset: 0

Author

Richard V. Scholtz, III, Sep 07 2013

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000771, A008277.

Represents the seventh column of results of A163626.

Programs

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

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 *)

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

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

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