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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.
24
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).
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.
Sequence in context: A307772 A225203 A136255 * A305320 A159813 A157409
KEYWORD
nonn,tabl,changed
AUTHOR
Omar E. Pol, Feb 04 2012
STATUS
approved