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

0,5

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)).

Links

Table of n, a(n) for n=0..63.

Example

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;

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]

Crossrefs

Sequence in context: A154372 A080416 A213166 * A099759 A350819 A072590

Adjacent sequences: A168616 A168617 A168618 * A168620 A168621 A168622

Keyword

nonn,tabl

Author

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

Status

approved