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 A131408 Repeated integer partitions or nested integer partitions. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 = 1..750 FORMULA a(1)=1, a(2)=2, 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 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 (VB) Sub test_A131408() Dim n As Long, Result As Long For n = 1 To 9 Result = A131408(n) Debug.Print n, Result Cells(3, 3 + n) = Result Next n End Sub Public Function A131408(n As Long) Dim imsgbox As Integer Dim i As Long, j As Long, Summe As Long If n = 0 Then A131408 = 0 Exit Function ElseIf n = 1 Then A131408 = 1 Exit Function ElseIf n = 2 Then A131408 = 2 Exit Function ElseIf n > 2 And n < 13 Then 'Summe = Bell(n) Summe = ZahlAllerPartitionen(n) For j = 2 To n - 1 'Summe = Summe + Stirling2(n, j) * A131408(j) Summe = Summe + ZahlPartitionen(n, j) * A131408(j) Next j Else imsgbox = MsgBox("Illegal input for argument *** n *** !", vbOKOnly, "A131408") End End If A131408 = Summe End Function Public Function ZahlAllerPartitionen(n As Long) Dim k As Long ZahlAllerPartitionen = 0 For k = 1 To n ZahlAllerPartitionen = ZahlAllerPartitionen + ZahlPartitionen(n, k) Next k End Function Sub TestZahlPartitonenInTeile() Dim n As Long, k As Long, Resultat As Long n = 8 k = 4 Resultat = ZahlPartitionen(n, k) Debug.Print "TestZahlPartitonen: n, k, Resultat:", n, k, Resultat End Sub Public Function ZahlPartitionen(n As Long, k As Long) ' compute recursively the number of partitions of n into k parts. Dim imsgbox As Integer If n > 2147483648# Or k > 2147483648# Then imsgbox = MsgBox("n and k need to be smaller than 2147483648 !", vbOKOnly, "ZahlPartitionen") End End If If (n < 0 Or k < 0) Then imsgbox = MsgBox("n and k need to be greater than 0 !", vbOKOnly, "ZahlPartitionen") End End If 'If k > n Then 'imsgbox = MsgBox("k needs to be <= n !", vbOKOnly, "ZahlPartitionen") 'End 'End If If k = 1 Then ZahlPartitionen = 1 Exit Function ElseIf k = n Then ZahlPartitionen = 1 Exit Function ElseIf k > n Then ZahlPartitionen = 0 Exit Function End If ZahlPartitionen = ZahlPartitionen(n - 1, k - 1) + ZahlPartitionen(n - k, k) End Function CROSSREFS Cf. A022811, A022818, A131407, A246828. Sequence in context: A080039 A265226 A299164 * A137917 A244099 A201371 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 STATUS approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)