A228913 - OEIS

%I #35 May 03 2017 18:49:54

%S 0,0,0,0,0,0,0,0,0,0,3628800,239500800,8821612800,239740300800,

%T 5368729766400,105006251750400,1858166876966400,30449278610150400,

%U 469614684719980800,6897777008118796800,97349279409046828800,1329165939158093836800,17651395149921751680000

%N a(n) = 11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1.

%C Calculates the eleventh 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="/A228913/b228913.txt">Table of n, a(n) for n = 0..960</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (66, -1925, 32670, -357423, 2637558, -13339535, 45995730, -105258076, 150917976, -120543840, 39916800).

%F G.f.: -3628800*x^10 / ((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)*(11*x-1)). - _Colin Barker_, Sep 20 2013

%F E.g.f.: Sum_{k=1..11} (-1)^(11-k)*binomial(11-1,k-1)*exp(k*x). - _Wolfdieter Lang_, May 03 2017

%t Table[10!*StirlingS2[n+1, 11], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)

%t CoefficientList[Series[-3628800*x^10 / ((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)*(11*x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 16 2014 *)

%t Table[11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1, {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 16 2014 *)

%t LinearRecurrence[{66,-1925,32670,-357423,2637558,-13339535,45995730,-105258076,150917976,-120543840,39916800},{0,0,0,0,0,0,0,0,0,0,3628800},30] (* _Harvey P. Dale_, Mar 20 2017 *)

%o (PARI) a(n)=11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1

%Y Eleventh column of results of A163626.

%Y Cf. A228910 (with more cf.s), A228911, A228912.

%K nonn,easy

%O 0,11

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

%E Offset corrected by _Vaclav Kotesovec_, Dec 16 2014