A364910 Number of integer partitions of 2n whose distinct parts sum to n.

1, 1, 1, 3, 3, 4, 12, 11, 19, 23, 54, 55, 103, 115, 178, 289, 389, 507, 757, 970, 1343, 2033, 2579, 3481, 4840, 6312, 8317, 10998, 15459, 19334, 26368, 33480, 44709, 56838, 74878, 93369, 128109, 157024, 206471, 258357, 338085, 417530, 544263, 669388, 859570, 1082758, 1367068

Offset

0,4

Comments

Also the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 91 terms from David A. Corneth)

Formula

a(n) = A116861(2n,n).

a(n) = A364916(n,n).

Example

The a(0) = 1 through a(7) = 11 partitions:
  ()  (11)  (22)  (33)     (44)      (55)       (66)         (77)
                  (2211)   (3311)    (3322)     (4422)       (4433)
                  (21111)  (311111)  (4411)     (5511)       (5522)
                                     (4111111)  (33321)      (6611)
                                                (42222)      (442211)
                                                (322221)     (4222211)
                                                (332211)     (4421111)
                                                (3222111)    (42221111)
                                                (3321111)    (422111111)
                                                (32211111)   (611111111)
                                                (51111111)   (4211111111)
                                                (321111111)
The a(0) = 1 through a(7) = 11 linear combinations:
  0  1*1  1*2  1*3      1*4      1*5      1*6          1*7
               0*2+3*1  0*3+4*1  0*4+5*1  0*4+3*2      0*6+7*1
               1*2+1*1  1*3+1*1  1*3+1*2  0*5+6*1      1*4+1*3
                                 1*4+1*1  1*4+1*2      1*5+1*2
                                          1*5+1*1      1*6+1*1
                                          0*3+0*2+6*1  0*4+0*2+7*1
                                          0*3+1*2+4*1  0*4+1*2+5*1
                                          0*3+2*2+2*1  0*4+2*2+3*1
                                          0*3+3*2+0*1  0*4+3*2+1*1
                                          1*3+0*2+3*1  1*4+0*2+3*1
                                          1*3+1*2+1*1  1*4+1*2+1*1
                                          2*3+0*2+0*1

Mathematica

Table[Length[Select[IntegerPartitions[2n], Total[Union[#]]==n&]], {n, 0, 15}]

Prog

(PARI) a(n) = {my(res = 0); forpart(p = 2*n, s = Set(p); if(vecsum(s) == n, res++)); res} \\ David A. Corneth, Aug 20 2023
(Python)
from sympy.utilities.iterables import partitions
def A364910(n): return sum(1 for d in partitions(n<<1, k=n) if sum(set(d))==n) # Chai Wah Wu, Sep 13 2023

Crossrefs

The case with no zero coefficients is A000009.

Central diagonal of A116861.

A version based on Heinz numbers is A364906.

Using all partitions (not just strict) we get A364907.

The version for compositions is A364908, strict A364909.

Main diagonal of A364916.

Using strict partitions of any number from 1 to n gives A365002.

These partitions have ranks A365003.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A323092 counts double-free partitions, ranks A320340.

Cf. A237113, A364350, A364839, A364911, A364912, A364914.

Sequence in context: A178043 A049925 A362107 * A291873 A118114 A366555

Adjacent sequences: A364907 A364908 A364909 * A364911 A364912 A364913

Keyword

nonn

Author

Gus Wiseman, Aug 16 2023

Extensions

More terms from David A. Corneth, Aug 20 2023

Status

approved