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A168619 Triangle T(n,k) read by rows with the coefficient [x^k] of the polynomial (x+1)^n + (2*n-3) *( (x+1)^n -x^n -1 ) in column k, row n.

%I A168619 #6 Jul 11 2012 11:26:11
%S A168619 1,1,1,1,4,1,1,12,12,1,1,24,36,24,1,1,40,80,80,40,1,1,60,150,200,150,
%T A168619 60,1,1,84,252,420,420,252,84,1,1,112,392,784,980,784,392,112,1,1,144,
%U A168619 576,1344,2016,2016,1344,576,144,1,1,180,810,2160,3780,4536,3780,2160,810
%N A168619 Triangle T(n,k) read by rows with the coefficient [x^k] of the polynomial (x+1)^n + (2*n-3) *( (x+1)^n -x^n -1 ) in column k, row n.
%C A168619 The term T(0,0) is defined to be 1.
%C A168619 Row sums are s(n) = 1, 2, 6, 26, 86, 242, 622, 1514, 3558, 8162, 18398,... (apparently with s(n) = 6*s(n-1) -13*s(n-2) +12*s(n-3)-4*s(n-4)).
%e A168619 1;
%e A168619 1, 1;
%e A168619 1, 4, 1;
%e A168619 1, 12, 12, 1;
%e A168619 1, 24, 36, 24, 1;
%e A168619 1, 40, 80, 80, 40, 1;
%e A168619 1, 60, 150, 200, 150, 60, 1;
%e A168619 1, 84, 252, 420, 420, 252, 84, 1;
%e A168619 1, 112, 392, 784, 980, 784, 392, 112, 1;
%e A168619 1, 144, 576, 1344, 2016, 2016, 1344, 576, 144, 1;
%e A168619 1, 180, 810, 2160, 3780, 4536, 3780, 2160, 810, 180, 1;
%p A168619 A168619 := proc(n,k)
%p A168619     if n = 0 then
%p A168619         1;
%p A168619     else
%p A168619         (1+x)^n + (2*n-3)*((1+x)^n-x^n-1) ;
%p A168619         coeftayl(%,x=0,k) ;
%p A168619     end if;
%p A168619 end proc: # _R. J. Mathar_, Jul 11 2012
%t A168619 p[x_, n_] := (x + 1)^n + If[n == 0, 0, (2*n - 3)]*((x + 1)^n - x^n - 1)
%t A168619 a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t A168619 Flatten[a]
%K A168619 nonn,tabl
%O A168619 0,5
%A A168619 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009