A194812 Square array read by antidiagonals: T(n,k) = number of parts of size k in the last section of the set of partitions of n.

1, 1, 0, 2, 1, 0, 3, 0, 0, 0, 5, 2, 1, 0, 0, 7, 1, 0, 0, 0, 0, 11, 4, 1, 1, 0, 0, 0, 15, 3, 2, 0, 0, 0, 0, 0, 22, 8, 2, 1, 1, 0, 0, 0, 0, 30, 7, 3, 1, 0, 0, 0, 0, 0, 0, 42, 15, 6, 3, 1, 1, 0, 0, 0, 0, 0, 56, 15, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 77, 27, 10

Offset

1,4

Comments

It appears that in the column k, starting in row n, the sum of k successive terms is equal to A000041(n-1).

Links

Table of n, a(n) for n=1..81.

Formula

It appears that A000041(n) = Sum_{j=1..k} T(n+j,k), n >= 0, k >= 1.

Example

Array begins:
.  1,  0,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  1,  1,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  2,  0,  1,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  3,  2,  0,  1, 0, 0, 0, 0, 0, 0, 0, 0,...
.  5,  1,  1,  0, 1, 0, 0, 0, 0, 0, 0, 0,...
.  7,  4,  2,  1, 0, 1, 0, 0, 0, 0, 0, 0,...
. 11,  3,  2,  1, 1, 0, 1, 0, 0, 0, 0, 0,...
. 15,  8,  3,  3, 1, 1, 0, 1, 0, 0, 0, 0,...
. 22,  7,  6,  2, 2, 1, 1, 0, 1, 0, 0, 0,...
. 30, 15,  6,  5, 3, 2, 1, 1, 0, 1, 0, 0,...
. 42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1, 0,...
. 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1,...
...
For n = 7, from the conjecture we have that p(n-1) = p(6) = 11 = 3+8 = 2+3+6 = 1+3+2+5 = 1+1+2+3+4 = 0+1+1+2+2+5, etc. where p(n) = A000041(n).

Crossrefs

Columns 1-4: A000041, A182712, A182713, A182714. Main triangle: A182703.

Cf. A066633, A135010, A138121, A138137.

Sequence in context: A307772 A225203 A136255 * A305320 A159813 A157409

Adjacent sequences: A194809 A194810 A194811 * A194813 A194814 A194815

Keyword

nonn,tabl

Author

Omar E. Pol, Feb 04 2012

Status

approved