This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364910 #24 Sep 14 2023 01:12:13
%S A364910 1,1,1,3,3,4,12,11,19,23,54,55,103,115,178,289,389,507,757,970,1343,
%T A364910 2033,2579,3481,4840,6312,8317,10998,15459,19334,26368,33480,44709,
%U A364910 56838,74878,93369,128109,157024,206471,258357,338085,417530,544263,669388,859570,1082758,1367068
%N A364910 Number of integer partitions of 2n whose distinct parts sum to n.
%C A364910 Also the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of n.
%H A364910 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 A364910 a(n) = A116861(2n,n).
%F A364910 a(n) = A364916(n,n).
%e A364910 The a(0) = 1 through a(7) = 11 partitions:
%e A364910 () (11) (22) (33) (44) (55) (66) (77)
%e A364910 (2211) (3311) (3322) (4422) (4433)
%e A364910 (21111) (311111) (4411) (5511) (5522)
%e A364910 (4111111) (33321) (6611)
%e A364910 (42222) (442211)
%e A364910 (322221) (4222211)
%e A364910 (332211) (4421111)
%e A364910 (3222111) (42221111)
%e A364910 (3321111) (422111111)
%e A364910 (32211111) (611111111)
%e A364910 (51111111) (4211111111)
%e A364910 (321111111)
%e A364910 The a(0) = 1 through a(7) = 11 linear combinations:
%e A364910 0 1*1 1*2 1*3 1*4 1*5 1*6 1*7
%e A364910 0*2+3*1 0*3+4*1 0*4+5*1 0*4+3*2 0*6+7*1
%e A364910 1*2+1*1 1*3+1*1 1*3+1*2 0*5+6*1 1*4+1*3
%e A364910 1*4+1*1 1*4+1*2 1*5+1*2
%e A364910 1*5+1*1 1*6+1*1
%e A364910 0*3+0*2+6*1 0*4+0*2+7*1
%e A364910 0*3+1*2+4*1 0*4+1*2+5*1
%e A364910 0*3+2*2+2*1 0*4+2*2+3*1
%e A364910 0*3+3*2+0*1 0*4+3*2+1*1
%e A364910 1*3+0*2+3*1 1*4+0*2+3*1
%e A364910 1*3+1*2+1*1 1*4+1*2+1*1
%e A364910 2*3+0*2+0*1
%t A364910 Table[Length[Select[IntegerPartitions[2n],Total[Union[#]]==n&]],{n,0,15}]
%o A364910 (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 A364910 (Python)
%o A364910 from sympy.utilities.iterables import partitions
%o A364910 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 A364910 The case with no zero coefficients is A000009.
%Y A364910 Central diagonal of A116861.
%Y A364910 A version based on Heinz numbers is A364906.
%Y A364910 Using all partitions (not just strict) we get A364907.
%Y A364910 The version for compositions is A364908, strict A364909.
%Y A364910 Main diagonal of A364916.
%Y A364910 Using strict partitions of any number from 1 to n gives A365002.
%Y A364910 These partitions have ranks A365003.
%Y A364910 A000041 counts integer partitions, strict A000009.
%Y A364910 A008284 counts partitions by length, strict A008289.
%Y A364910 A323092 counts double-free partitions, ranks A320340.
%Y A364910 Cf. A237113, A364350, A364839, A364911, A364912, A364914.
%K A364910 nonn
%O A364910 0,4
%A A364910 _Gus Wiseman_, Aug 16 2023
%E A364910 More terms from _David A. Corneth_, Aug 20 2023