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 A239955 #31 Aug 18 2025 16:26:19
%S A239955 0,0,0,0,1,2,4,7,12,17,27,38,54,75,104,137,187,245,322,418,542,691,
%T A239955 887,1121,1417,1777,2228,2767,3441,4247,5235,6424,7871,9594,11688,
%U A239955 14173,17168,20723,24979,30008,36010,43085,51479,61357,73032,86718,102852,121718
%N A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).
%C A239955 From _Gus Wiseman_, Jun 26 2022: (Start)
%C A239955 Also the number of partitions of n with at least one gap, i.e., partitions whose parts do not form a contiguous interval. These partitions are ranked by A073492. For example, the a(0) = 0 through a(8) = 12 partitions are:
%C A239955 . . . . (31) (41) (42) (52) (53)
%C A239955 (311) (51) (61) (62)
%C A239955 (411) (331) (71)
%C A239955 (3111) (421) (422)
%C A239955 (511) (431)
%C A239955 (4111) (521)
%C A239955 (31111) (611)
%C A239955 (3311)
%C A239955 (4211)
%C A239955 (5111)
%C A239955 (41111)
%C A239955 (311111)
%C A239955 Also the number of non-constant partitions of n with a repeated non-maximal part, ranked by A065201. The a(0) = 0 through a(8) = 12 partitions are:
%C A239955 . . . . (211) (311) (411) (322) (422)
%C A239955 (2111) (2211) (511) (611)
%C A239955 (3111) (3211) (3221)
%C A239955 (21111) (4111) (3311)
%C A239955 (22111) (4211)
%C A239955 (31111) (5111)
%C A239955 (211111) (22211)
%C A239955 (32111)
%C A239955 (41111)
%C A239955 (221111)
%C A239955 (311111)
%C A239955 (2111111)
%C A239955 (End)
%H A239955 Alois P. Heinz, <a href="/A239955/b239955.txt">Table of n, a(n) for n = 0..10000</a> (first 201 terms from John Tyler Rascoe)
%F A239955 a(n) = A000041(n) - A034296(n).
%F A239955 G.f.: Sum_{i>1} q^i/(q;q)_{i-1} * Sum_{j=1..i-1} (q^2;q^2)_{j-2} where (a;q)_k = Product_{i>=0..k} (1-a*q^i). - _John Tyler Rascoe_, Aug 16 2025
%e A239955 a(6) counts these 4 partitions: 51, 42, 411, 3111.
%p A239955 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A239955 `if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
%p A239955 end:
%p A239955 a:= n-> combinat[numbpart](n)-add(b(n, k), k=0..n):
%p A239955 seq(a(n), n=0..47); # _Alois P. Heinz_, Aug 18 2025
%t A239955 z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
%t A239955 Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (*A239954*)
%t A239955 Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
%t A239955 Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
%t A239955 Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (*A034296*)
%t A239955 Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
%t A239955 (* second program *)
%t A239955 Table[Length[Select[IntegerPartitions[n],Min@@Differences[#]<-1&]],{n,0,30}] (* _Gus Wiseman_, Jun 26 2022 *)
%o A239955 (PARI)
%o A239955 qs(a,q,n) = {prod(k=0,n,1-a*q^k)}
%o A239955 A_q(N) = {if(N<4, vector(N+1,i,0), my(q='q+O('q^(N-2)), g= sum(i=2,N+1, q^i/qs(q,q,i-1)*sum(j=1,i-1, q^(2*j)*qs(q^2,q^2,j-2)))); concat([0,0,0,0], Vec(g)))} \\ _John Tyler Rascoe_, Aug 16 2025
%Y A239955 Cf. A239954, A239956, A239958.
%Y A239955 The complement is counted by A034296 (strict A137793), ranked by A073491.
%Y A239955 These partitions are ranked by A073492, conjugate A065201.
%Y A239955 Applying the condition to the conjugate gives A350839, ranked by A350841.
%Y A239955 A000041 counts integer partitions, strict A000009.
%Y A239955 A090858 counts partitions with a single hole, ranked by A325284.
%Y A239955 A116931 counts partitions with differences != -1, strict A003114.
%Y A239955 A116932 counts partitions with differences != -1 or -2, strict A025157.
%Y A239955 Cf. A000070, A001227, A008284, A144300, A183558, A321440.
%K A239955 nonn,easy,changed
%O A239955 0,6
%A A239955 _Clark Kimberling_, Mar 30 2014