A213385 a(n) = number of refinements of the partition n^1. (Formerly N0320)

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

1,2

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.

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.

Example

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.

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 *)

Crossrefs

Cf. A002846, A213242, A213427.

Sequence in context: A289470 A161746 A368564 * A045629 A034731 A216435

Adjacent sequences: A213382 A213383 A213384 * A213386 A213387 A213388

Keyword

nonn

Author

N. J. A. Sloane, Jun 10 2012

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

Status

approved