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A131408 Repeated integer partitions or nested integer partitions.

%I A131408 #28 Mar 02 2024 12:28:30
%S A131408 1,1,2,5,14,35,95,248,668,1781,4799,12890,34766,93647,252635,681272,
%T A131408 1838135,4958738,13379885,36100214,97409045,262833314,709207394,
%U A131408 1913652308,5163654671,13933178390,37596275726,101446960109,273737216768,738632652929,1993073801930
%N A131408 Repeated integer partitions or nested integer partitions.
%C A131408 See A131407 for the labeled case (with much more explanation).
%C A131408 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
%H A131408 Alois P. Heinz, <a href="/A131408/b131408.txt">Table of n, a(n) for n = 0..2321</a>
%H A131408 Thomas Wieder, <a href="/A131408/a131408.txt">Visual Basic Program</a>
%F A131408 a(n) = A000041(n) + Sum_{i=2..n-1} A008284(n,i)*a(i).
%F A131408 a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399866618837633..., c = 0.232635324064951140265176690908381464098550827908380222089145... . - _Vaclav Kotesovec_, Sep 04 2014
%e A131408 Let denote * an unlabeled element. Then a(n=3)=5 because we have [ *,*,* ], [ *, * ][ * ], [[ *,* ]][[ * ]], [[ *,* ][ * ]], [ * ][ * ][ * ].
%e A131408 From _Gus Wiseman_, Jul 20 2018: (Start)
%e A131408 The a(4) = 14 sequences of integer partitions:
%e A131408   (4), (31), (22), (211),
%e A131408   (4)(1), (31)(2), (22)(2), (211)(3), (211)(21),
%e A131408   (31)(2)(1), (22)(2)(1), (211)(3)(1), (211)(21)(2),
%e A131408   (211)(21)(2)(1).
%e A131408 (End)
%p A131408 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
%p A131408 # second Maple program:
%p A131408 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A131408       b(n, i-1) + b(n-i, min(n-i, i)))
%p A131408     end:
%p A131408 a:= proc(n) option remember; b(n$2)+
%p A131408       add(b(n-i, min(n-i, i))*a(i), i=2..n-1)
%p A131408     end:
%p A131408 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 03 2020
%t A131408 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 *)
%t A131408 roo[n_]:=If[n==1,{{{1}}},Join@@Cases[Most[IntegerPartitions[n]],y_:>Prepend[(Prepend[#,y]&/@roo[Length[y]]),{y}]]];
%t A131408 Table[Length[roo[n]],{n,10}] (* _Gus Wiseman_, Jul 20 2018 *)
%o A131408 (Visual Basic) ' See Wieder link.
%Y A131408 Cf. A022811, A022818, A131407, A246828.
%K A131408 nonn
%O A131408 0,3
%A A131408 _Thomas Wieder_, Jul 09 2007
%E A131408 Edited and extended by _R. J. Mathar_, Aug 07 2008
%E A131408 a(0)=1 prepended and edited by _Alois P. Heinz_, Sep 03 2020