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A131408 Repeated integer partitions or nested integer partitions. 6
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).

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 [ *,*,* ], [ *, * ][ * ], [[ *,* ]][[ * ]], [[ *,* ][ * ]], [ * ][ * ][ * ].

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-Fran├žois Alcover, Feb 02 2017 *)

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. 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 February 20 12:30 EST 2018. Contains 299379 sequences. (Running on oeis4.)