%I #32 Jan 26 2025 16:29:31
%S 1,1,1,1,1,1,1,2,1,1,2,2,1,3,2,2,1,3,4,2,1,4,5,4,2,1,4,6,5,4,1,5,8,8,
%T 5,4,1,5,10,10,8,5,1,6,11,14,10,8,5,1,6,14,16,16,10,8,1,7,16,22,20,16,
%U 10,8,1,7,18,26,27,20,16,10,1,8,21,32,34,31
%N Table read by rows: number of complete partitions of n with largest part = k.
%C See A126796 for definition of complete partitions;
%C A126796(n) = sum of n-th row;
%C also T(n,floor((n+1)/2)) = A126796(floor(n/2)).
%H Reinhard Zumkeller, <a href="/A261036/b261036.txt">Rows n = 1..200 of triangle, flattened</a>
%H SeungKyung Park, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-4/park.pdf">Complete Partitions</a>, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.
%F According to the Park link, Theorem 3.7, p. 357f:
%F Let D_k(n) be the number of complete partitions of a positive integer n with largest part exactly k.
%F D_0(n) = 0 for all n, D_k(0) = 0 for all k, D_1(n)=1 for n>0, and for k>1:
%F D_k(n) = D_(k-1)(n-1) + D_k(n-k) if n >= 3*k-1, D_(k-1)(n-1) if 2*k-1 <= n <= 3*k-2, 0 if 1 <= n <= 2*k-2.
%F In the following, T(n,k) = D_k(n).
%e T(8,2) = #{1+1+1+1+1+1+2, 1+1+1+1+2+2, 1+1+2+2+2} = 3;
%e T(8,3) = #{1+1+1+1+1+3, 1+1+1+2+3, 1+1+3+3, 1+2+2+3} = 4;
%e T(8,4) = #{1+1+1+1+4, 1+1+2+4} = 2;
%e T(9,2) = #{+11+1+1+1+1+1+2, 1+1+1+1+1+2+2, 1+1+1+2+2+2, 1+2+2+2+2} = 4;
%e T(9,3) = #{1+1+1+1+1+1+3, 1+1+1+1+2+3, 1+1+1+3+3, 1+1+2+2+3, 3,3,2,1} = 5;
%e T(9,4) = #{1+1+1+1+1+4, 1+1+1+2+4, 1+1+3+4, 1+2+2+4} = 4;
%e T(9,5) = #{1+1+1+1+5, 1+1+2+2+5} = 2.
%e . -----------------------------------------------
%e . n | T(n,k), k = 1 .. [(n+1)/2] | A126796(n)
%e . ----+------------------------------+-----------
%e . 1 | 1 | 1
%e . 2 | 1 | 1
%e . 3 | 1 1 | 2
%e . 4 | 1 1 | 2
%e . 5 | 1 2 1 | 4
%e . 6 | 1 2 2 | 5
%e . 7 | 1 3 2 2 | 8
%e . 8 | 1 3 4 2 | 10
%e . 9 | 1 4 5 4 2 | 16
%e . 10 | 1 4 6 5 4 | 20
%e . 11 | 1 5 8 8 5 4 | 31
%e . 12 | 1 5 10 10 8 5 | 39
%e . 13 | 1 6 11 14 10 8 5 | 55
%e . 14 | 1 6 14 16 16 10 8 | 71
%e . 15 | 1 7 16 22 20 16 10 8 | 100
%e . 16 | 1 7 18 26 27 20 16 10 | 125
%e . 17 | 1 8 21 32 34 31 20 16 10 | 173
%e . 18 | 1 8 24 37 42 39 31 20 16 | 218
%e . 19 | 1 9 26 46 53 50 39 31 20 16 | 291
%e . 20 | 1 9 30 52 66 63 55 39 31 20 | 366
%t d[k_, n_] := d[k, n] = Which[n == 0 || k == 0, 0, k == 1, 1, n >= 3 k - 1, d[k - 1, n - 1] + d[k, n - k], 2 k - 1 <= n <= 3 k - 2, d[k - 1, n - 1], True, 0]; Table[d[k, n], {n, 17}, {k, Floor[(n + 1)/2]}] // Flatten (* _Michael De Vlieger_, Jul 13 2017 *)
%o (Haskell)
%o import Data.MemoCombinators (memo2, integral, Memo)
%o a261036 n k = a261036_tabf !! (n-1) !! (k-1)
%o a261036_row n = a261036_tabf !! (n-1)
%o a261036_tabf = zipWith (map . flip dMemo) [1..] a122197_tabf where
%o dMemo = memo2 integral integral d
%o d 0 _ = 0
%o d _ 0 = 0
%o d 1 _ = 1
%o d k n | n <= 2 * k - 2 = 0
%o | n <= 3 * k - 2 = dMemo (k - 1) (n - 1)
%o | otherwise = dMemo (k - 1) (n - 1) + dMemo k (n - k)
%Y Cf. A008619 (row lengths), A126796 (row sums).
%Y Cf. A122197.
%K nonn,tabf,look
%O 1,8
%A _Reinhard Zumkeller_, Aug 08 2015