cp's OEIS Frontend

A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

%I A079126 #21 Feb 16 2025 08:32:48
%S A079126 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 A079126 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 A079126 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 A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.
%C A079126 T(n,n) = A000009(n), right side of the triangle;
%C A079126 T(n,k)=0 for n>0 and k < A002024(n); T(prime(n),n) = A067953(n) for n>0.
%H A079126 Alois P. Heinz, <a href="/A079126/b079126.txt">Rows n = 0..140, flattened</a>
%H A079126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>.
%F A079126 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 A079126 T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - _Vladeta Jovovic_, Aug 07 2003
%e A079126 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 A079126 Triangle T(n,k) begins:
%e A079126 1;
%e A079126 0, 1;
%e A079126 0, 0, 1;
%e A079126 0, 0, 1, 2;
%e A079126 0, 0, 0, 1 ,2;
%e A079126 0, 0, 0, 1, 2, 3;
%e A079126 0, 0, 0, 1, 2, 3, 4;
%e A079126 0, 0, 0, 0, 2, 3, 4, 5;
%e A079126 0, 0, 0, 0, 1, 3, 4, 5,  6;
%e A079126 0, 0, 0, 0, 1, 3, 5, 6,  7,  8;
%e A079126 0, 0, 0, 0, 1, 3, 5, 7,  8,  9, 10;
%e A079126 0, 0, 0, 0, 0, 2, 5, 7,  9, 10, 11, 12;
%e A079126 0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15; ...
%p A079126 T:= proc(n, i) option remember; `if`(n=0, 1,
%p A079126       `if`(i<1, 0, T(n, i-1)+`if`(i>n, 0, T(n-i, i-1))))
%p A079126     end:
%p A079126 seq(seq(T(n,k), k=0..n), n=0..20);  # _Alois P. Heinz_, May 11 2015
%t A079126 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 A079126 Cf. A000009, A079122, A035294, A079124, A079125.
%Y A079126 Differs from A026840 in having extra zeros at the ends of the rows.
%K A079126 nonn,tabl
%O A079126 0,10
%A A079126 _Reinhard Zumkeller_, Dec 27 2002