A228912 - OEIS

%I #38 Oct 11 2017 05:10:07

%S 0,0,0,0,0,0,0,0,0,362880,19958400,618710400,14270256000,273158645760,

%T 4595022432000,70309810771200,1000944296352000,13467262000832640,

%U 173201547619900800,2147373231974006400,25832386565857872000,303056981918271947520,3481253462769108364800

%N 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.

%C 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)].

%H Seiichi Manyama, <a href="/A228912/b228912.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

%F 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

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

%t Table[9!*StirlingS2[n+1, 10], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)

%t 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 *)

%t 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_ *)

%o (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

%Y Tenth column of results of A163626.

%Y Essentially 362880*A049435.

%Y Cf. A228910 (with more crossrefs), A228911.

%K nonn,easy

%O 0,10

%A _Richard V. Scholtz, III_, Sep 07 2013

%E Offset corrected by _Vaclav Kotesovec_, Dec 16 2014