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

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

Offset: 1

Author

Alford Arnold, Jun 08 2003

Keywords

• nonn
• tabf

Comments

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

Examples

			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
...
		

Crossrefs

Cf. A049597, A033638.

Programs

Maple

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

Mathematica

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 *)

Extensions

More terms from Emeric Deutsch, May 15 2004