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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

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

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

EXAMPLE

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)

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 *)

PROG

(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

CROSSREFS

Cf. A026807.

Cf. A002260, A060016.

Sequence in context: A177994 A179285 A079211 * A117975 A143258 A027199

Adjacent sequences:  A026832 A026833 A026834 * A026836 A026837 A026838

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)