A079126 - OEIS

%I #20 Nov 04 2024 12:27:34

%S 1,0,1,0,0,1,0,0,1,2,0,0,0,1,2,0,0,0,1,2,3,0,0,0,1,2,3,4,0,0,0,0,2,3,

%T 4,5,0,0,0,0,1,3,4,5,6,0,0,0,0,1,3,5,6,7,8,0,0,0,0,1,3,5,7,8,9,10,0,0,

%U 0,0,0,2,5,7,9,10,11,12,0,0,0,0,0,2,5,8,10,12,13,14,15,0,0,0,0,0,1,4,8,11,13,15,16,17,18

%N Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

%C T(n,n) = A000009(n), right side of the triangle;

%C T(n,k)=0 for n>0 and k < A002024(n); T(prime(n),n) = A067953(n) for n>0.

%H Alois P. Heinz, <a href="/A079126/b079126.txt">Rows n = 0..140, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>.

%F T(n,k) = b(0,n,k), where b(m,n,k) = 1+sum(b(i,j,k): m<i<j<k and i+j=n).

%F T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - _Vladeta Jovovic_, Aug 07 2003

%e The seven partitions of n=5 are {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1} and {1,1,1,1,1}. Only two of them ({4,1} and {3,2}) have distinct parts <= 4, so T(5,4) = 2.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, 0, 1, 2;

%e 0, 0, 0, 1 ,2;

%e 0, 0, 0, 1, 2, 3;

%e 0, 0, 0, 1, 2, 3, 4;

%e 0, 0, 0, 0, 2, 3, 4, 5;

%e 0, 0, 0, 0, 1, 3, 4, 5,  6;

%e 0, 0, 0, 0, 1, 3, 5, 6,  7,  8;

%e 0, 0, 0, 0, 1, 3, 5, 7,  8,  9, 10;

%e 0, 0, 0, 0, 0, 2, 5, 7,  9, 10, 11, 12;

%e 0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15; ...

%p T:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, T(n, i-1)+`if`(i>n, 0, T(n-i, i-1))))

%p     end:

%p seq(seq(T(n,k), k=0..n), n=0..20);  # _Alois P. Heinz_, May 11 2015

%t T[n_, i_] := T[n, i] = If[n==0, 1, If[i<1, 0, T[n, i-1] + If[i>n, 0, T[n-i, i-1]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Jun 30 2015, after _Alois P. Heinz_ *)

%Y Cf. A000009, A079122, A035294, A079124, A079125.

%Y Differs from A026840 in having extra zeros at the ends of the rows.

%K nonn,tabl

%O 0,10

%A _Reinhard Zumkeller_, Dec 27 2002