login
A249543
Square array T(m,n) of integer partitions with m addends n+1, read by antidiagonals.
4
1, 2, 3, 4, 9, 7, 6, 20, 26, 15, 10, 40, 72, 68, 30, 14, 75, 171, 220, 159, 56, 21, 133, 379, 614, 603, 352, 101, 29, 229, 786, 1559, 1928, 1525, 732, 176, 41, 383, 1568, 3700, 5564, 5534, 3618, 1465, 297
OFFSET
1,2
COMMENTS
T(m,n) is the integer partition with m times the addend n+1 (and no other non-one addends) given as index number of A194602.
The entries in the array A249544 are also in the sequence A194602. This array T contains the index numbers of A194602 corresponding to the entries of that array: A194602(T(m,n)) = A249544(m,n).
Row 1 is A000065, column 1 is A058695 (both with shifted index).
FORMULA
A194602(T(m,n)) = A249544(m,n).
T(1,n) = A000065(n+1) = p(n+1) - 1.
T(2,n) = p(2*(n+1)) - 2.
T(3,n) = p(3*(n+1)) - floor((n+1)/2) - 3.
T(m,1) = A058695(m-1) = p(2n-1).
p is the sequence of partition numbers A000041. (See "Identities for A194602" link.)
EXAMPLE
T(5,2) = 159.
A194602(159) = 14043. (So A249544(5,2) = 14043.)
14043 in binary is 11011011011011. That corresponds to the integer partition with 5 times the addend 3. (See row 159 in "Table for A194602" link.)
Array begins:
n 1 2 3 4 5 6 7 8 9
m
1 1 2 4 6 10 14 21 29 41
2 3 9 20 40 75 133 229 383
3 7 26 72 171 379 786 1568
4 15 68 220 614 1559 3700
5 30 159 603 1928 5564
6 56 352 1525 5534
7 101 732 3618
8 176 1465
9 297
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tilman Piesk, Oct 31 2014
STATUS
approved