A219207 Triangle, read by rows, where T(n,k) = binomial(n,k)^(k+1) for n>=0, k=0..n.

1, 1, 1, 1, 4, 1, 1, 9, 27, 1, 1, 16, 216, 256, 1, 1, 25, 1000, 10000, 3125, 1, 1, 36, 3375, 160000, 759375, 46656, 1, 1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1, 1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1, 1, 81, 46656

Offset: 0

Author

Paul D. Hanna, Nov 14 2012

Keywords

• nonn
• tabl

Comments

Maximal term in row n is asymptotically in position k = r*n, where r = A220359 = 0.70350607643... is a root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Nov 15 2012

Examples

			Triangle of coefficients C(n,k)^(k+1) begins:
1;
1, 1;
1, 4, 1;
1, 9, 27, 1;
1, 16, 216, 256, 1;
1, 25, 1000, 10000, 3125, 1;
1, 36, 3375, 160000, 759375, 46656, 1;
1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1;
1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1; ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
3,-4,1;
-73,99,-27,1;
18055,-24496,6696,-256,1;
-55694851,75563975,-20656000,790000,-3125,1; - _R. J. Mathar_, Mar 22 2013
		

Links

• Paul D. Hanna, Rows n = 0..45, flattened.

Crossrefs

Cf. A184731, A219206, A184730.

Programs

Mathematica

Table[Binomial[n,k]^(k+1),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2016 *)

PARI

{T(n,k)=binomial(n,k)^(k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Row sums equal A184731.