A249543 Square array T(m,n) of integer partitions with m addends n+1, read by antidiagonals.

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

Author

Tilman Piesk, Oct 31 2014

Keywords

• nonn
• tabl

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).

Examples

			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
		

Links

• Tilman Piesk, First 113 rows of the triangle, flattened
• Tilman Piesk, Table for A194602
• Li-yao Xia, Identities for A194602

Crossrefs

Cf. A194602, A249544, A000065, A058695, A000041.

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.)