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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.
2

%I #16 Oct 15 2012 22:15:17

%S 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,

%T 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,

%U 20478,37251,7546,387,13,1,1,0,1,14,3311,312511,1749868,688733,41377,895,16,1,1,0

%N 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.

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

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

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%e T(6,2) = #{partitions of binomial(6,2)=15 into parts i with 2<i<=6}

%e = #{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;

%e T(6,3) = #{partitions of binomial(6,3)=20 into parts i with 3<i<=6}

%e = #{6+6+4+4, 6+5+5+4, 5+5+5+5, 4+4+4++4+4} = 4.

%o (Haskell)

%o a090824 n k = a090824_tabl !! n !! k

%o a090824_row n = a090824_tabl !! n

%o a090824_tabl = zipWith (zipWith p)

%o (map (\x -> map (`enumFromTo` x) [1..x+1]) [0..]) a007318_tabl

%o where p _ 0 = 1

%o p [] _ = 0

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Oct 10 2012

%Y Cf. A000041, A007318.

%K nonn,tabl

%O 0,12

%A _Reinhard Zumkeller_, Feb 16 2004

%E Offset and example corrected by _Reinhard Zumkeller_, Oct 10 2012