OFFSET
0,3
COMMENTS
See A131407 for the labeled case (with much more explanation).
Also the number of sequences of distinct integer partitions (y_1, ..., y_k), containing no partitions of the form (111..1) other than (1), such that sum(y_1) = n and length(y_i) = sum(y_{i+1}) for all i = 1, ..., k-1. - Gus Wiseman, Jul 20 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2321
Thomas Wieder, Visual Basic Program
FORMULA
a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399866618837633..., c = 0.232635324064951140265176690908381464098550827908380222089145... . - Vaclav Kotesovec, Sep 04 2014
EXAMPLE
Let denote * an unlabeled element. Then a(n=3)=5 because we have [ *,*,* ], [ *, * ][ * ], [[ *,* ]][[ * ]], [[ *,* ][ * ]], [ * ][ * ][ * ].
From Gus Wiseman, Jul 20 2018: (Start)
The a(4) = 14 sequences of integer partitions:
(4), (31), (22), (211),
(4)(1), (31)(2), (22)(2), (211)(3), (211)(21),
(31)(2)(1), (22)(2)(1), (211)(3)(1), (211)(21)(2),
(211)(21)(2)(1).
(End)
MAPLE
A000041 := proc(n) combinat[numbpart](n) ; end: A008284 := proc(n, k) if k = 1 or k = n then 1; elif k > n then 0 ; else procname(n-1, k-1)+procname(n-k, k) ; fi ; end: A131408 := proc(n) option remember; local i ; if n <= 2 then n; else A000041(n)+add(A008284(n, i)*procname(i), i=2..n-1) ; fi ; end: for n from 1 to 40 do printf("%d, ", A131408(n)) ; od: # R. J. Mathar, Aug 07 2008
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) + b(n-i, min(n-i, i)))
end:
a:= proc(n) option remember; b(n$2)+
add(b(n-i, min(n-i, i))*a(i), i=2..n-1)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 03 2020
MATHEMATICA
t[_, 1] = 1; t[n_, k_] /; 1 <= k <= n := t[n, k] = Sum[t[n-i, k-1], {i, 1, n-1}] - Sum[t[n-i, k], {i, 1, k-1}]; t[_, _] = 0; a[1]=1; a[2]=2; a[n_] := a[n] = PartitionsP[n] + Sum[t[n, i]*a[i], {i, 2, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 02 2017 *)
roo[n_]:=If[n==1, {{{1}}}, Join@@Cases[Most[IntegerPartitions[n]], y_:>Prepend[(Prepend[#, y]&/@roo[Length[y]]), {y}]]];
Table[Length[roo[n]], {n, 10}] (* Gus Wiseman, Jul 20 2018 *)
PROG
(Visual Basic) ' See Wieder link.
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Wieder, Jul 09 2007
EXTENSIONS
Edited and extended by R. J. Mathar, Aug 07 2008
a(0)=1 prepended and edited by Alois P. Heinz, Sep 03 2020
STATUS
approved