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A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

%I A382864 #23 Apr 07 2025 09:26:29
%S A382864 1,0,1,0,1,0,1,1,0,1,1,0,1,2,0,1,2,1,0,1,3,1,0,1,3,2,0,1,4,3,0,1,4,4,
%T A382864 1,0,1,5,5,1,0,1,5,7,2,0,1,6,8,3,0,1,6,10,5,0,1,7,12,6,1,0,1,7,14,9,1,
%U A382864 0,1,8,16,11,2,0,1,8,19,15,3,0,1,9,21,18,5,0,1,9,24,23,7
%N A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).
%H A382864 Seiichi Manyama, <a href="/A382864/b382864.txt">Rows n = 0..481, flattened</a>
%F A382864 G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).
%F A382864 T(n,k) = |A292047(n,k)| = |A292049(n,k)|.
%e A382864 First few rows are:
%e A382864   1;
%e A382864   0, 1;
%e A382864   0, 1;
%e A382864   0, 1, 1;
%e A382864   0, 1, 1;
%e A382864   0, 1, 2;
%e A382864   0, 1, 2,  1;
%e A382864   0, 1, 3,  1;
%e A382864   0, 1, 3,  2;
%e A382864   0, 1, 4,  3;
%e A382864   0, 1, 4,  4, 1;
%e A382864   0, 1, 5,  5, 1;
%e A382864   0, 1, 5,  7, 2;
%e A382864   0, 1, 6,  8, 3;
%e A382864   0, 1, 6, 10, 5;
%e A382864   0, 1, 7, 12, 6, 1;
%e A382864   ...
%Y A382864 Row sums give A000009.
%Y A382864 Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
%Y A382864 Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.
%K A382864 nonn,tabf
%O A382864 0,14
%A A382864 _Seiichi Manyama_, Apr 07 2025