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 A228910 a(n) = 8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1. 9
 0, 0, 0, 0, 0, 0, 0, 5040, 181440, 3780000, 59875200, 801496080, 9574044480, 105398092800, 1091804313600, 10794490827120, 102896614941120, 952741767650400, 8617145057539200, 76461500619902160, 667855517349303360, 5757691363157764800, 49099453300298016000, 414884142077935345200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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)]. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1107 Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320). FORMULA a(n) = 5040 * S2(n+1,8), n>=0. 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 E.g.f.: Sum_{k=1..8} (-1)^(8-k)*binomial(8-1,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}, Table[ -row[n], {n, 7, 23}] [[All, 8]]] (* Jean-François Alcover, Dec 16 2014 *) Table[7!*StirlingS2[n + 1, 8], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *) 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 *) 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 *) PROG (PARI) a(n)=8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1. (PARI) for(n=0, 30, print1(5040*stirling(n+1, 8, 2), ", ")) \\ G. C. Greubel, Nov 19 2017 (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 CROSSREFS Cf. A008277, A049434. The eighth column of results of A163626. Cf. A000225, A028243, A028244, A028245, A032180, A228909. Sequence in context: A213877 A135456 A254080 * A258419 A179062 A342075 Adjacent sequences:  A228907 A228908 A228909 * A228911 A228912 A228913 KEYWORD nonn,easy AUTHOR Richard V. Scholtz, III, Sep 07 2013 EXTENSIONS Offset corrected by Vaclav Kotesovec, Dec 16 2014 STATUS approved

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Last modified January 24 21:02 EST 2022. Contains 350565 sequences. (Running on oeis4.)