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

1,5

Links

Table of n, a(n) for n=1..82.

Index entries for sequences related to compositions

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

Example

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).

Prog

(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

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.

Sequence in context: A303810 A052249 A030528 * A089263 A348951 A369815

Adjacent sequences: A077224 A077225 A077226 * A077228 A077229 A077230

Keyword

nonn,tabl

Author

Henry Bottomley, Oct 29 2002

Status

approved