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)
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
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