A228912 a(n) = 10^n - 9*9^n + 36*8^n - 84*7^n + 126*6^n - 126*5^n + 84*4^n - 36*3^n + 9*2^n - 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 19958400, 618710400, 14270256000, 273158645760, 4595022432000, 70309810771200, 1000944296352000, 13467262000832640, 173201547619900800, 2147373231974006400, 25832386565857872000, 303056981918271947520, 3481253462769108364800

Offset

0,10

Comments

Calculates the tenth 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..1000

Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Formula

G.f.: 362880*x^9 / ((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)*(10*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..10} (-1)^(10-k)*binomial(10-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Mathematica

Table[9!*StirlingS2[n+1, 10], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[362880*x^9 / ((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)*(10*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)

Prog

(PARI) a(n)=10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1

Crossrefs

Tenth column of results of A163626.

Essentially 362880*A049435.

Cf. A228910 (with more crossrefs), A228911.

Sequence in context: A179967 A133360 A254082 * A213871 A179064 A246197

Adjacent sequences: A228909 A228910 A228911 * A228913 A228914 A228915

Keyword

nonn,easy

Author

Richard V. Scholtz, III, Sep 07 2013

Extensions

Offset corrected by Vaclav Kotesovec, Dec 16 2014

Status

approved