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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.

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