A026835 Triangular array read by rows: T(n,k) = number of partitions of n into distinct parts in which every part is >=k, for k=1,2,...,n.

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, 18, 10, 6, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 22, 12, 7, 4, 3, 2, 1, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n,1)=A000009(n), T(n,2)=A025147(n) for n>1, T(n,3)=A025148(n) for n>2, T(n,4)=A025149(n) for n>3.

A219922(n) = smallest number of row containing n. - Reinhard Zumkeller, Dec 01 2012

Examples

			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)
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Cf. A026807.

Cf. A002260, A060016.

Programs

Haskell

import Data.List (tails)
a026835 n k = a026835_tabl !! (n-1) !! (k-1)
a026835_row n = a026835_tabl !! (n-1)
a026835_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0      _ = 1
         p _     [] = 0
         p m (k:ks) = if m < k then 0 else p (m - k) ks + p m ks
-- Reinhard Zumkeller, Dec 01 2012

Mathematica

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 *)

Formula

G.f.: Sum_{k>=1} (y^k*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 25 2003

T(n, k) = 1 + Sum(T(i, j): i>=j>k and i+j=n+1). - Reinhard Zumkeller, Jan 01 2003

T(n, k) > 1 iff 2*k < n. - Reinhard Zumkeller, Jan 01 2003