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%I #21 Sep 06 2020 11:42:47
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C A219922(n) = smallest number of row containing n. - _Reinhard Zumkeller_, Dec 01 2012
%H Reinhard Zumkeller, <a href="/A026835/b026835.txt">Rows n = 1..120 of triangle, flattened</a>
%F G.f.: Sum_{k>=1} (y^k*(-1+Product_{i>=k} (1+x^i))). - _Vladeta Jovovic_, Aug 25 2003
%F T(n, k) = 1 + Sum(T(i, j): i>=j>k and i+j=n+1). - _Reinhard Zumkeller_, Jan 01 2003
%F T(n, k) > 1 iff 2*k < n. - _Reinhard Zumkeller_, Jan 01 2003
%e From _Michael De Vlieger_, Aug 03 2020: (Start)
%e Table begins:
%e 1
%e 1 1
%e 2 1 1
%e 2 1 1 1
%e 3 2 1 1 1
%e 4 2 1 1 1 1
%e 5 3 2 1 1 1 1
%e 6 3 2 1 1 1 1 1
%e 8 5 3 2 1 1 1 1 1
%e 10 5 3 2 1 1 1 1 1 1
%e 12 7 4 3 2 1 1 1 1 1 1
%e 15 8 5 3 2 1 1 1 1 1 1 1
%e ... (End)
%t 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 *)
%o (Haskell)
%o import Data.List (tails)
%o a026835 n k = a026835_tabl !! (n-1) !! (k-1)
%o a026835_row n = a026835_tabl !! (n-1)
%o a026835_tabl = map
%o (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
%o where p 0 _ = 1
%o p _ [] = 0
%o p m (k:ks) = if m < k then 0 else p (m - k) ks + p m ks
%o -- _Reinhard Zumkeller_, Dec 01 2012
%Y Cf. A026807.
%Y Cf. A002260, A060016.
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_
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