1,2

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 (FrankTAW(AT)Netscape.net), May 27 2008

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

...

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)

Cf. A049597, A033638.

Sequence in context: A117716 A097150 A087165 this_sequence A023133 A026280 A115994

Adjacent sequences: A083477 A083478 A083479 this_sequence A083481 A083482 A083483

nonn,tabf

Alford Arnold (Alford1940(AT)Aol.com), Jun 08 2003

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2004