A069713 As a square array T(n,k) by antidiagonals, number of ways of partitioning k into up to n parts each no more than 5, or into up to 5 parts each no more than n; as a triangle t(n,k), number of ways of partitioning n into exactly k parts each no more than 6 (i.e., of arranging k indistinguishable standard dice to produce a total of n).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 0, 3, 4, 3, 2, 1, 1, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 2, 6, 6, 5, 3, 2, 1, 1, 0, 0, 2, 6, 8, 7, 5, 3, 2, 1, 1, 0, 0, 1, 6, 9, 9, 7, 5, 3, 2, 1, 1, 0, 0, 1, 6, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0, 0, 0, 5, 11, 14, 12, 10, 7, 5

Offset: 0

Author

Henry Bottomley, Apr 01 2002

Keywords

• nonn
• tabl

Examples

			As square array, rows start: 1,0,0,0,0,0,...; 1,1,1,1,1,1,...; 1,1,2,2,3,3,...; 1,1,2,3,4,5,...; 1,1,2,3,5,6,...; 1,1,2,3,5,7,...; etc. As triangle, rows start: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,1,1; 0,1,2,2,1,1; 0,1,3,3,2,1,1; etc. T(3,7)=6 since 7 can be written as 5+2, 5+1+1, 4+3, 4+2+1, 3+3+1, 3+2+2; or alternatively as 2+2+1+1+1, 3+1+1+1, 2+2+2+1, 3+2+1+1, 3+2+2, 3+3+1. t(10,3)=6 since 10 can be written as 6+3+1, 6+2+2, 5+4+1, 5+3+2, 4+4+2, 4+3+3.
		

Crossrefs

Cf. A061676 for a similar triangle, though with distinguishable dice (and a different offset). Antidiagonal sums of T(n, k), i.e., row sums (over k) of t(n, k), are A001402. First 22 terms are same as A068914 (see formula).

Formula

If k<6 T(n,k) = A068914(n,k). T(n,k) = T(n,5n-k); t(n,k) = t(7n-k,k). T(floor(5n/2),n) = t(n,floor(7n/2)) = A001975(n).