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

0,13

Links

Table of n, a(n) for n=0..100.

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).

Example

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).

Sequence in context: A029400 A370173 A344612 * A319453 A072233 A264391

Adjacent sequences: A069710 A069711 A069712 * A069714 A069715 A069716

Keyword

nonn,tabl

Author

Henry Bottomley, Apr 01 2002

Status

approved