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Number of integer partitions of 2n whose distinct parts sum to n.
18

%I #24 Sep 14 2023 01:12:13

%S 1,1,1,3,3,4,12,11,19,23,54,55,103,115,178,289,389,507,757,970,1343,

%T 2033,2579,3481,4840,6312,8317,10998,15459,19334,26368,33480,44709,

%U 56838,74878,93369,128109,157024,206471,258357,338085,417530,544263,669388,859570,1082758,1367068

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

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

%H Alois P. Heinz, <a href="/A364910/b364910.txt">Table of n, a(n) for n = 0..500</a> (first 91 terms from David A. Corneth)

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

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

%e The a(0) = 1 through a(7) = 11 partitions:

%e () (11) (22) (33) (44) (55) (66) (77)

%e (2211) (3311) (3322) (4422) (4433)

%e (21111) (311111) (4411) (5511) (5522)

%e (4111111) (33321) (6611)

%e (42222) (442211)

%e (322221) (4222211)

%e (332211) (4421111)

%e (3222111) (42221111)

%e (3321111) (422111111)

%e (32211111) (611111111)

%e (51111111) (4211111111)

%e (321111111)

%e The a(0) = 1 through a(7) = 11 linear combinations:

%e 0 1*1 1*2 1*3 1*4 1*5 1*6 1*7

%e 0*2+3*1 0*3+4*1 0*4+5*1 0*4+3*2 0*6+7*1

%e 1*2+1*1 1*3+1*1 1*3+1*2 0*5+6*1 1*4+1*3

%e 1*4+1*1 1*4+1*2 1*5+1*2

%e 1*5+1*1 1*6+1*1

%e 0*3+0*2+6*1 0*4+0*2+7*1

%e 0*3+1*2+4*1 0*4+1*2+5*1

%e 0*3+2*2+2*1 0*4+2*2+3*1

%e 0*3+3*2+0*1 0*4+3*2+1*1

%e 1*3+0*2+3*1 1*4+0*2+3*1

%e 1*3+1*2+1*1 1*4+1*2+1*1

%e 2*3+0*2+0*1

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

%o (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

%o (Python)

%o from sympy.utilities.iterables import partitions

%o 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

%Y The case with no zero coefficients is A000009.

%Y Central diagonal of A116861.

%Y A version based on Heinz numbers is A364906.

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

%Y The version for compositions is A364908, strict A364909.

%Y Main diagonal of A364916.

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

%Y These partitions have ranks A365003.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A323092 counts double-free partitions, ranks A320340.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 16 2023

%E More terms from _David A. Corneth_, Aug 20 2023