editing

approved

t(n,k+1)<=t(n,k)<=t(n+1,k); t(n,k)>1 iff 0<=2*(k+1)<n.

Álvar Ibeas, <a href="/A079211/b079211.txt">First 100 rows, flattened</a>

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

Reinhard Zumkeller, <a href="/A079211/a079211.txt">Notes on initial terms</a>

t(n, k) = 1 + Sum(t(i, j): i>j>k and i+j=n).

From Michael De Vlieger, Aug 03 2020: (Start)

Table begins:

1

1 1

2 1 1

2 1 1 1

3 2 1 1 1

4 2 1 1 1 1

5 3 2 1 1 1 1

6 3 2 1 1 1 1 1

8 5 3 2 1 1 1 1 1

10 5 3 2 1 1 1 1 1 1

12 7 4 3 2 1 1 1 1 1 1

15 8 5 3 2 1 1 1 1 1 1 1

... (End)

Nest[Function[{T, n, r}, Append[T, Table[1 + Total[T[[##]] & @@@ Select[r, #[[-1]] > k + 1 &]], {k, 0, n}]]] @@ {#1, #2, Transpose[1 + {#2 - #3, #3}]} & @@ {#1, #2, Range[Ceiling[#2/2] - 1]} & @@ {#, Length@ #} &, {{1}}, 12] // Flatten (* Michael De Vlieger, Aug 03 2020 *)

A000009(n)=t(n, 0), left border of the triangle, A025147(n)=t(n, 1) for n>1.

Reinhard Zumkeller, Jan 01 2003

Corrected by Álvar Ibeas, Sep 06 2020

approved

editing

proposed

approved

editing

proposed

Álvar Ibeas: Thanks! I’ll move the formula too.

Triangle t(n,k) of numbers of partitions of n into distinct numbers larger than k, 0<=k<n.

Duplicate of A026835.

nonn,tabl,changed

dead

proposed

editing

Joerg Arndt: Took care of it, also of A026835.

editing

proposed

Triangle t(n,k) of numbers of partitions of n into distinct numbers notlarger lessthan k, 0<=k<=<n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 1, 8, 5, 3, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 12, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 15, 8, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 18, 10, 6, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 22, 12, 7, 4

0,7

1,4

Álvar Ibeas, <a href="/A079211/b079211.txt">First 100 rows, flattened</a>

t(n, k) = 1 + Sum(t(i, j): i>j>k and i+j=n), 0<=k<=n.).

Corrected by Álvar Ibeas, Sep 06 2020

Álvar Ibeas: After the correction, it coincides with A026835. Should them be merged?

proposed

editing

editing

proposed

Álvar Ibeas: Note that the formula (tuning it according to the chosen definition) follows from the sequence name, classifying the enumerated partitions by smallest part. I think a name involving partitions is better than a formula.

N. J. A. Sloane: We seem to have several different suggestions here.  I am moving it to the editing stack.

From Michael De Vlieger, Aug 03 2020: (Start)Table begins:

Table begins:

1 1 2 1 1 2 1 1 1 3 2 1 1 1 4 2 1 1 1 1 5 3 2 1 1 1 1 6 3 2 1 1 1 1 1 8 5 3 2 1 1 1 1 1 10 5 3 2 1 1 1 1 1 1 12 7 4 3 2 1 1 1 1 1 1 15 8 5 3 2 1 1 1 1 1 1 1 ...(End)

1 1

2 1 1

2 1 1 1

3 2 1 1 1

4 2 1 1 1 1

5 3 2 1 1 1 1

6 3 2 1 1 1 1 1

8 5 3 2 1 1 1 1 1

10 5 3 2 1 1 1 1 1 1

12 7 4 3 2 1 1 1 1 1 1

15 8 5 3 2 1 1 1 1 1 1 1

... (End)