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A212551 Number of partitions T(n,k) of n containing at least one other part m-k if m is the largest part; triangle T(n,k), n>=0, 0<=k<=n.

%I A212551 #20 Dec 13 2016 20:29:57
%S A212551 1,0,0,1,0,0,1,1,0,0,2,1,1,0,0,2,3,1,1,0,0,4,3,3,1,1,0,0,4,6,4,3,1,1,
%T A212551 0,0,7,7,7,4,3,1,1,0,0,8,11,9,8,4,3,1,1,0,0,12,13,15,10,8,4,3,1,1,0,0,
%U A212551 14,20,18,17,11,8,4,3,1,1,0,0
%N A212551 Number of partitions T(n,k) of n containing at least one other part m-k if m is the largest part; triangle T(n,k), n>=0, 0<=k<=n.
%C A212551 Reversed rows converge to A024786.
%H A212551 Alois P. Heinz, <a href="/A212551/b212551.txt">Rows n = 0..140, flattened</a>
%H A212551 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_delta">Kronecker delta</a>
%F A212551 G.f. of column k: delta_{0,k} + Sum_{i>0} x^(2*i+k) / Product_{j=1..k+i} (1-x^j), where delta is the Kronecker delta.
%e A212551 T(4,0) = 2: [1,1,1,1], [2,2].
%e A212551 T(4,1) = 1: [2,1,1].
%e A212551 T(5,1) = 3: [2,1,1,1], [2,2,1], [3,2].
%e A212551 T(6,2) = 3: [3,1,1,1], [3,2,1], [4,2].
%e A212551 T(7,2) = 4: [3,1,1,1,1], [3,2,1,1], [3,3,1], [4,2,1].
%e A212551 T(8,4) = 3: [5,1,1,1], [5,2,1], [6,2].
%e A212551 Triangle T(n,k) begins:
%e A212551 1;
%e A212551 0, 0;
%e A212551 1, 0, 0;
%e A212551 1, 1, 0, 0;
%e A212551 2, 1, 1, 0, 0;
%e A212551 2, 3, 1, 1, 0, 0;
%e A212551 4, 3, 3, 1, 1, 0, 0;
%e A212551 4, 6, 4, 3, 1, 1, 0, 0;
%e A212551 7, 7, 7, 4, 3, 1, 1, 0, 0;
%p A212551 b:= proc(n, i) option remember;
%p A212551       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212551     end:
%p A212551 T:= (n, k)-> `if`(n=0 and k=0, 1,
%p A212551     add(b(n-2*m-k, min(n-2*m-k, m+k)), m=1..(n-k)/2)):
%p A212551 seq(seq(T(n, k), k=0..n), n=0..14);
%t A212551 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i > n, 0, b[n-i, i]]]; t[n_, k_] := If[n == 0 && k == 0, 1, Sum[b[n-2*m-k, Min[n-2*m-k, m+k]], {m, 1, (n-k)/2}]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 13 2013, translated from Maple *)
%Y A212551 Columns k=0-10 give: A002865, A083751(n+1), A119907, A212543, A212544, A212545, A212546, A212547, A212548, A212549, A212550.
%Y A212551 Row sums give A000070(n-2) for n>1.
%Y A212551 Cf. A024786.
%K A212551 nonn,tabl
%O A212551 0,11
%A A212551 _Alois P. Heinz_, May 20 2012