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

1,9

Links

Table of n, a(n) for n=1..78.

Formula

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

Example

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.

Prog

(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(""))

Crossrefs

Cf. A221834, A000169, A077227.

Sequence in context: A055334 A106237 A071675 * A319082 A034365 A103778

Adjacent sequences: A221830 A221831 A221832 * A221834 A221835 A221836

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 26 2013

Status

approved