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

1, 2, 3, 7, 15, 43, 131, 468, 1776, 7559, 34022, 166749, 853823, 4682358, 26720781, 161074458, 1004485751, 6576974188, 44322716809, 311440019349, 2247888977510, 16819336465164, 128915407382036, 1021269823516449, 8261243728564640, 68848043979970646

Offset: 1

Author

N. J. A. Sloane, Jun 10 2012

Keywords

• nonn

Comments

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.

Examples

			Referring to the ranked poset L(5) shown in the example in A002846, there are 15 paths that start at ooooo:
end point / number of paths
ooooo / 1
o oooo / 1
oo ooo / 1
o o ooo / 2
o oo oo / 2
o o o oo / 4
o o o o o / 4
Total a(5) = 15.
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• Olivier Gérard, The ranked posets L(2),...,L(8)
• R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence labeled H.

Crossrefs

Cf. A002846, A213242, A213427.

Programs

Maple

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(l[n]=1 and {l[1..n-1][]} minus {0}={}, 1,
      add(`if`(l[i]=0, 0, add(`if`(l[j]=0 or i=j and l[j]<2, 0,
      b([seq(`if`(t>n, 0, l[t])-`if`(t=i and t=j, 2, `if`(t=i or t=j,
      1, `if`(t=i+j, -1, 0))), t=1..max(n, i+j))])), j=i..n)), i=1..n))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 11 2012

Mathematica

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 *)

Extensions

Definition clarified by David Applegate, Jun 10 2012

More terms from Alois P. Heinz, Jun 11 2012

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016