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%I #25 Feb 24 2024 10:05:21
%S 1,0,0,1,0,1,1,1,0,2,1,1,2,1,2,3,2,2,5,2,4,5,5,4,8,5,7,9,8,8,13,10,11,
%T 16,13,15,20,18,18,27,21,26,31,30,30,43,34,42,49,48,48,65,56,65,76,74,
%U 77,97,88,98,117,111,119,143,137,146,175,165,182,208
%N Number of strict integer partitions of n where all parts have neighbors.
%C A part x has a neighbor if either x - 1 or x + 1 is a part.
%H Alois P. Heinz, <a href="/A356606/b356606.txt">Table of n, a(n) for n = 0..5000</a> (first 301 terms from John Tyler Rascoe)
%H John Tyler Rascoe, <a href="/A356606/a356606.py.txt">Python program</a>
%F G.f.: 1 + Sum_{i>0} A(x,i), where A(x,i) = x^((2*i)+1) * G(x,i+1) for i > 0, is the g.f. for partitions of this kind with least part i, and G(x,k) = 1 + x^(k+1) * G(x,k+1) + Sum_{m>=0} x^(2*(k+m)+5) * G(x,m+k+3). - _John Tyler Rascoe_, Feb 16 2024
%e The a(n) partitions for n = 0, 1, 3, 9, 15, 18, 20, 24 (A = 10, B = 11):
%e () . (21) (54) (87) (765) (7643) (987)
%e (432) (654) (6543) (8732) (8754)
%e (54321) (7632) (9821) (9843)
%e (8721) (65432) (A932)
%e (65421) (BA21)
%e (87432)
%e (87621)
%e (765321)
%t Table[Length[Select[IntegerPartitions[n], Function[ptn,UnsameQ@@ptn&&And@@Table[MemberQ[ptn,x-1]||MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%o (Python) # see linked program
%Y This is the strict case of A355393 and A355394.
%Y The complement is counted by A356607, non-strict A356235 and A356236.
%Y A000041 counts integer partitions, strict A000009.
%Y A000837 counts relatively prime partitions, ranked by A289509.
%Y A007690 counts partitions with no singletons, complement A183558.
%Y Cf. A137921, A325160, A328171, A328172, A328187, A328220, A328221, A356237.
%K nonn
%O 0,10
%A _Gus Wiseman_, Aug 24 2022