A090824 Triangle read by rows: T(n,k) = number of partitions of binomial(n,k) into parts greater than k and not greater than n, 0<=k<=n.

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 7, 4, 1, 1, 0, 1, 4, 20, 19, 5, 1, 1, 0, 1, 7, 67, 159, 57, 8, 1, 1, 0, 1, 8, 238, 1607, 1163, 157, 9, 1, 1, 0, 1, 12, 868, 20478, 37251, 7546, 387, 13, 1, 1, 0, 1, 14, 3311, 312511, 1749868, 688733, 41377, 895, 16, 1, 1, 0

Offset: 0

Author

Reinhard Zumkeller, Feb 16 2004

Keywords

• nonn
• tabl

Comments

n>0: T(n,0) = 1, T(n,1) = A002865(n);

T(n,n-2) = 1 for n>1; T(n,n-1) = 1 for n>0; T(n,n) = 0.

Examples

			T(6,2) = #{partitions of binomial(6,2)=15 into parts i with 2<i<=6}
= #{6+6+3, 6+5+4, 6+3+3+3, 5+5+5, 5+4+3+3, 4+4+4+3, 3+3+3+3+3} = 7;
T(6,3) = #{partitions of binomial(6,3)=20 into parts i with 3<i<=6}
= #{6+6+4+4, 6+5+5+4, 5+5+5+5, 4+4+4++4+4} = 4.
		

Links

• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000041, A007318.

Programs

Haskell

a090824 n k = a090824_tabl !! n !! k
a090824_row n = a090824_tabl !! n
a090824_tabl = zipWith (zipWith p)
   (map (\x -> map (`enumFromTo` x) [1..x+1]) [0..]) a007318_tabl
   where p _          0 = 1
         p []         _ = 0
         p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Oct 10 2012

Extensions

Offset and example corrected by Reinhard Zumkeller, Oct 10 2012