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 A237978 #17 Feb 25 2025 20:44:31
%S A237978 0,0,0,0,0,0,0,1,0,0,1,2,1,3,1,10,6,6,13,15,20,30,30,64,61,85,99,153,
%T A237978 170,184,271,359,416,539,631,799,971,1199,1433,1803,2153,2692,3173,
%U A237978 3882,4544,5610,6663,8090,9565,11467,13590,16305,19184,22933,26973
%N A237978 Number of partitions of n having depth 3; see Comments.
%C A237978 Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P),..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is introduced here as the depth of P. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.
%e A237978 a(14) = 3 counts these partitions: 64211, 632111, 433211.
%e A237978 Successive applications of f to the first of these partitions are indicated by 64211 -> 6422 -> 644 -> 86.
%t A237978 z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
%t A237978 Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
%t A237978 Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
%t A237978 Table[Count[c[n], 3], {n, 1, z}] (* this sequence *)
%t A237978 (* _Peter J. C. Moses_, Feb 19 2014 *)
%Y A237978 Cf. A237685, A237750, A366063, A000009, A000041.
%K A237978 nonn,easy
%O A237978 1,12
%A A237978 _Clark Kimberling_, Feb 19 2014