A131408 Repeated integer partitions or nested integer partitions.

1, 1, 2, 5, 14, 35, 95, 248, 668, 1781, 4799, 12890, 34766, 93647, 252635, 681272, 1838135, 4958738, 13379885, 36100214, 97409045, 262833314, 709207394, 1913652308, 5163654671, 13933178390, 37596275726, 101446960109, 273737216768, 738632652929, 1993073801930

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) = A000041(n) + Sum_{i=2..n-1} A008284(n,i)*a(i).

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

Cf. A022811, A022818, A131407, A246828.

Sequence in context: A357835 A369591 A299164 * A137917 A360031 A244099

Adjacent sequences: A131405 A131406 A131407 * A131409 A131410 A131411

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