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A083480
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Compress the triangular array A049597 by suppressing zero entries and reversing the order of each row.
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7
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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, 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, 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
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refs;
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OFFSET
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1,2
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COMMENTS
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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.
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
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LINKS
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EXAMPLE
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The table begins:
1
2
3
4 1
5 2
6 3 2
7 4 4
8 5 6 3
9 6 8 6 1
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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