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A083480 Compress the triangular array A049597 by suppressing zero entries and reversing the order of each row.

%I A083480 #14 Feb 19 2015 08:38:58
%S A083480 1,2,3,4,1,5,2,6,3,2,7,4,4,8,5,6,3,9,6,8,6,1,10,7,10,9,6,11,8,12,12,
%T A083480 11,2,12,9,14,15,16,9,2,13,10,16,18,21,16,7,14,11,18,21,26,23,18,4,15,
%U A083480 12,20,24,31,30,29,12,3,16,13,22,27,36,37,40,27,12,1,17,14,24,30,41,44,51
%N A083480 Compress the triangular array A049597 by suppressing zero entries and reversing the order of each row.
%C A083480 Row sums => A000041. Diagonals are sums of Gaussian polynomials (which then sum to powers of two). The number of entries on each row is conjectured to conform to: 0 1 1 1 2 2 3 3 4 5 5 6 7 7 8 9 10 10 11 12 13 13 14 15 16 17 17 ... a sequence which stutters after values 0,1,2,4,6,9,12,16,...A002620.
%C A083480 Regarding the first element of the sequence as T(1,0), it appears that this is the number of partitions of n with k elements not in the first hook; i.e., with n - (max part size) - (number of parts) + 1 = k. If this is correct, we have T(n,0) = n and for k > 0, T(n,k) = sum_{j >= max(0,2k-n+2)} j * T(k,j). This is equivalent to T(n,k) = T(n-1,k) + sum_{j >= max(0,2k-n+2)} T(k,j) and thus to T(n,k) = 2* T(n-1,k) - T(n-2,k) + T(k,2k-n+2) [taking T(n,k) = 0 if k < 0]. It also implies the correctness of the conjecture about the row lengths. - _Franklin T. Adams-Watters_, May 27 2008
%e A083480 The table begins:
%e A083480 1
%e A083480 2
%e A083480 3
%e A083480 4 1
%e A083480 5 2
%e A083480 6 3 2
%e A083480 7 4 4
%e A083480 8 5 6 3
%e A083480 9 6 8 6 1
%e A083480 ...
%p A083480 a:=n->sort(simplify(sum(product((1-q^i),i=n-r+1..n)/product((1-q^j),j=1..r), r=0..n))):T := proc(n,k) if k=n then n+1 elif k>n then 0 else coeff(a(k),q^(n-k)) fi end: b:=proc(n,k) if T(n,n-k)>0 then T(n,n-k) else fi end:seq(seq(b(n,k),k=0..n+1),n=0..20); # _Emeric Deutsch_, May 15 2004
%t A083480 a[n_] := Sum[Product[1-q^i, {i, n-r+1, n}]/Product[1-q^j, {j, 1, r}], {r, 0, n}] // Simplify; T [n_, k_] := Which[k == n, n+1, k>n, 0, True, Coefficient[a[k], q^(n - k)]]; Table[Table[T[n, k], {k, n, 0, -1}] // DeleteCases[#, 0]&, {n, 0,  21}] // Flatten (* _Jean-François Alcover_, Feb 19 2015, after Maple *)
%Y A083480 Cf. A049597, A033638.
%K A083480 nonn,tabf
%O A083480 1,2
%A A083480 _Alford Arnold_, Jun 08 2003
%E A083480 More terms from _Emeric Deutsch_, May 15 2004