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A213385 a(n) = number of refinements of the partition n^1.

%I A213385 N0320 #42 Mar 22 2017 08:34:38
%S A213385 1,2,3,7,15,43,131,468,1776,7559,34022,166749,853823,4682358,26720781,
%T A213385 161074458,1004485751,6576974188,44322716809,311440019349,
%U A213385 2247888977510,16819336465164,128915407382036,1021269823516449,8261243728564640,68848043979970646
%N A213385 a(n) = number of refinements of the partition n^1.
%C A213385 Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at n^1 and end at a node in the poset.
%D A213385 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A213385 Alois P. Heinz, <a href="/A213385/b213385.txt">Table of n, a(n) for n = 1..50</a>
%H A213385 Olivier Gérard, <a href="/A002846/a002846.png">The ranked posets L(2),...,L(8)</a>
%H A213385 R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: <a href="/A002572/a002572.jpg">front</a>, <a href="/A002572/a002572_1.jpg">back</a> [Annotated scanned copy, with permission] See sequence labeled H.
%e A213385 Referring to the ranked poset L(5) shown in the example in A002846, there are 15 paths that start at ooooo:
%e A213385 end point / number of paths
%e A213385 ooooo / 1
%e A213385 o oooo / 1
%e A213385 oo ooo / 1
%e A213385 o o ooo / 2
%e A213385 o oo oo / 2
%e A213385 o o o oo / 4
%e A213385 o o o o o / 4
%e A213385 Total a(5) = 15.
%p A213385 b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
%p A213385       `if`(l[n]=1 and {l[1..n-1][]} minus {0}={}, 1,
%p A213385       add(`if`(l[i]=0, 0, add(`if`(l[j]=0 or i=j and l[j]<2, 0,
%p A213385       b([seq(`if`(t>n, 0, l[t])-`if`(t=i and t=j, 2, `if`(t=i or t=j,
%p A213385       1, `if`(t=i+j, -1, 0))), t=1..max(n, i+j))])), j=i..n)), i=1..n))
%p A213385     end:
%p A213385 g:= proc(n, i, l)
%p A213385       `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
%p A213385       `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
%p A213385     end:
%p A213385 a:= n-> g(n, n, []):
%p A213385 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jun 11 2012
%t A213385 b[l_List] := b[l] = Module[{n, i, j, t}, n = Length[l]; If[l[[n]] == 1 && Union[ l[[1 ;; n-1]]] ~Complement~ {0} == {}, 1, Sum[If[l[[i]] == 0, 0,  Sum[If[l[[j]] == 0 || i == j && l[[j]]<2, 0, b[Table[If[t>n, 0, l[[t]]] - Which[t == i && t == j, 2, t == i || t == j, 1, t == i+j, -1, True, 0], {t, 1, Max[n, i+j]}]]], {j, i, n}] ], {i, 1, n}]]]; g[n_, i_, l_List] := If[n == 0 && i == 0, b[l], If[i == 1, b[ Join[{n}, l]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Join[{j}, l]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Feb 26 2015, after _Alois P. Heinz_ *)
%Y A213385 Cf. A002846, A213242, A213427.
%K A213385 nonn
%O A213385 1,2
%A A213385 _N. J. A. Sloane_, Jun 10 2012
%E A213385 Definition clarified by _David Applegate_, Jun 10 2012
%E A213385 More terms from _Alois P. Heinz_, Jun 11 2012
%E A213385 Edited by _Alois P. Heinz_ at the suggestion of _Gus Wiseman_, May 02 2016