A197184 - OEIS

%I #10 Mar 30 2012 17:40:29

%S 1,-1,1,-1,-1,1,7,-2,-1,1,-13,12,-3,-1,1,-17,-22,18,-4,-1,1,199,-45,

%T -35,25,-5,-1,1,-605,465,-84,-53,33,-6,-1,1,-225,-1449,910,-133,-77,

%U 42,-7,-1,1,11703,-864,-3094,1594,-190,-108,52,-8,-1,1,-59317,33780,-1380,-6027,2583,-252,-147,63,-9,-1,1,83143,-179398,78567,-771,-10899,3948,-315,-195,75,-10,-1,1,991671,271073,-461978,159115,2882,-18546,5764,-374,-253,88,-11,-1,1

%N Triangle of polynomial coefficients of the polynomial factors defined in A074051.

%C The triangle T(n,k), 0<=k<n, shows the coefficients [x^k] of the polynomial p_n(x) which distributes sum_{i=1..m} i^n*(i+1)! = A074052(n) + A074051(n)*sum_{i=1..m} (i+1)! + p_n(m) *(m+2)!.

%F A074052(n) + 2*A074051(n) + 6*p_n(1) = 2. - R. J. Mathar, Oct 13 2011

%F (x+2)*p_n(x)-p_n(x-1) = x^n-A074051(n). - R. J. Mathar, Oct 13 2011

%F Conjectures on p_n(x)= sum_{k=0..n-1} T(n,k)*x^k:

%F T(n,n-1) = 1.

%F T(n,n-2) = -1.

%F T(n,n-3) = -(n-2).

%F T(n,n-4) = A055998(n-2).

%F T(n,n-5) = -(n-2)*(n^2-4*n+21)/6.

%F T(n,n-6) = (n-5)*(n-2)*(n^2-19*n-24)/24.

%e 1;   1

%e -1,1;  -1+x

%e -1,-1,1;  -1-x+x^2

%e 7,-2,-1,1;  7-2*x-x^2+x^3

%e -13,12,-3,-1,1;  -13+12*x-3*x^2-x^3+x^4

%e -17,-22,18,-4,-1,1;   -17-22*x+18*x^2-4*x^3-x^4+x^5

%K sign,tabl

%O 1,7

%A _R. J. Mathar_, Oct 11 2011