A221833 Triangle, read by rows, where T(n,k) = [x^n] x^k*(1-x^k)^(k-1) / (1-x)^(k-1) for n>=k>=1.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 10, 10, 5, 1, 0, 0, 0, 12, 20, 15, 6, 1, 0, 0, 0, 12, 35, 35, 21, 7, 1, 0, 0, 0, 10, 52, 70, 56, 28, 8, 1, 0, 0, 0, 6, 68, 126, 126, 84, 36, 9, 1, 0, 0, 0, 3, 80, 205, 252, 210, 120, 45, 10, 1

Offset: 1

Author

Paul D. Hanna, Jan 26 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, 0, 2, 1;
0, 0, 3, 3, 1;
0, 0, 2, 6, 4, 1;
0, 0, 1, 10, 10, 5, 1;
0, 0, 0, 12, 20, 15, 6, 1;
0, 0, 0, 12, 35, 35, 21, 7, 1;
0, 0, 0, 10, 52, 70, 56, 28, 8, 1;
0, 0, 0, 6, 68, 126, 126, 84, 36, 9, 1;
0, 0, 0, 3, 80, 205, 252, 210, 120, 45, 10, 1;
0, 0, 0, 1, 85, 305, 462, 462, 330, 165, 55, 11, 1;
0, 0, 0, 0, 80, 420, 786, 924, 792, 495, 220, 66, 12, 1; ...
in which column sums equal: A000169(k) = k^(k-1) for k>=1.
		

Crossrefs

Cf. A221834, A000169, A077227.

Programs

PARI

{T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^(k-1),n-k)}
for(n=1,12,for(k=1,n,print1(T(n,k),", "));print(""))

Formula

G.f. of column k = x^k * ( (1-x^k)/(1-x) )^(k-1).