A077227 Triangle of compositions of n into exactly k parts each no more than k.

1, 0, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 6, 4, 1, 0, 0, 7, 10, 5, 1, 0, 0, 6, 20, 15, 6, 1, 0, 0, 3, 31, 35, 21, 7, 1, 0, 0, 1, 40, 70, 56, 28, 8, 1, 0, 0, 0, 44, 121, 126, 84, 36, 9, 1, 0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1, 0, 0, 0, 31, 255, 456, 462, 330, 165, 55, 11, 1, 0, 0, 0, 20

Offset: 1

Author

Henry Bottomley, Oct 29 2002

Keywords

• nonn
• tabl

Examples

			T(6,3)=7 since 6 can be written as 1+2+3, 1+3+2, 2+1+3, 2+2+2, 2+3+1, 3+1+2, or 3+2+1.
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 1, 3, 1;
0, 0, 6, 4, 1;
0, 0, 7, 10, 5, 1;
0, 0, 6, 20, 15, 6, 1;
0, 0, 3, 31, 35, 21, 7, 1;
0, 0, 1, 40, 70, 56, 28, 8, 1;
0, 0, 0, 44, 121, 126, 84, 36, 9, 1;
0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1; ...
where column sums are k^k (A000312).
		

Links

• Index entries for sequences related to compositions

Crossrefs

Column sums are A000312. Row sums are A077229. Central diagonal is A000984 offset. Right hand side is right hand side of A007318. Cf. A077228.

Programs

PARI

T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^k,n-k)
for(n=1,12,for(k=1,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jan 25 2013

Formula

T(n, k) = A077228(n, k) - A077228(n-1, k).

If n>=k^2, T(n, k) = 0. If k<=n<2k, T(n, k) = C(n-1, k-1).

G.f. of column k is: x^k*(1-x^k)^k/(1-x)^k for k>=1. - Paul D. Hanna, Jan 25 2013