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.

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, 60, 1, 1, 84, 252, 420, 420, 252, 84, 1, 1, 112, 392, 784, 980, 784, 392, 112, 1, 1, 144, 576, 1344, 2016, 2016, 1344, 576, 144, 1, 1, 180, 810, 2160, 3780, 4536, 3780, 2160, 810

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

• nonn
• tabl

Comments

The term T(0,0) is defined to be 1.

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)).

Examples

			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, 60, 1;
1, 84, 252, 420, 420, 252, 84, 1;
1, 112, 392, 784, 980, 784, 392, 112, 1;
1, 144, 576, 1344, 2016, 2016, 1344, 576, 144, 1;
1, 180, 810, 2160, 3780, 4536, 3780, 2160, 810, 180, 1;
		

Programs

Maple

A168619 := proc(n,k)
    if n = 0 then
        1;
    else
        (1+x)^n + (2*n-3)*((1+x)^n-x^n-1) ;
        coeftayl(%,x=0,k) ;
    end if;
end proc: # R. J. Mathar, Jul 11 2012

Mathematica

p[x_, n_] := (x + 1)^n + If[n == 0, 0, (2*n - 3)]*((x + 1)^n - x^n - 1)
a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a]