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A180174 Triangle read by rows of the numbers C(n,k) of k-subsets of a quadratically populated n-multiset M. 1
1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 5, 7, 9, 10, 10, 10, 10, 10, 9, 7, 5, 3, 1, 1, 4, 9, 16, 25, 35, 45, 55, 65, 75, 84, 91, 96, 99, 100, 100, 100, 99, 96, 91, 84, 75, 65, 55, 45, 35, 25, 16, 9, 4, 1, 1, 5, 14, 30, 55, 90, 135, 190, 255, 330, 414, 505, 601, 700, 800, 900, 1000, 1099 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The multiplicity m(i) of the i-th element with 1 <= i <= n is m(i)=i^2.

Thus M=[1,2,2,2,2,...,i^2 x i,...,n^2 x n].

Row sum is equal to A028361.

Column for k=2 is equal to AA000096.

Column for k=3 is equal to AA005581.

Column for k=4 is equal to AA005582.

The number of coefficients C(n,k) for given n is equal to A056520.

LINKS

Table of n, a(n) for n=0..72.

FORMULA

C(0,0) = 0.

C(n,k) = sum_{j=(k-LS+1)}^{k} C(n-1,j).

for n > 0 and k=1,...,LR with LS = n^2+1 and LR = n*(n+1)*(2*n+1)/6.

C(n,k) = C(n,LR-k).

EXAMPLE

For n=4 one has M=[1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4].

For k=7 we have 55 subsets from M:

[1, 2, 2, 3, 3, 4, 4], [1, 2, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 4, 4],

[1, 2, 2, 3, 4, 4, 4], [1, 2, 2, 3, 3, 3, 4], [1, 2, 2, 2, 3, 4, 4],

[1, 2, 2, 2, 3, 3, 4], [2, 2, 3, 3, 4, 4, 4], [2, 2, 3, 3, 3, 4, 4],

[2, 2, 2, 3, 3, 4, 4], [1, 2, 2, 2, 3, 3, 3], [1, 2, 2, 2, 4, 4, 4],

[1, 3, 3, 3, 4, 4, 4], [2, 3, 3, 3, 4, 4, 4], [2, 2, 2, 3, 4, 4, 4],

[2, 2, 2, 3, 3, 3, 4], [1, 2, 3, 4, 4, 4, 4], [1, 2, 3, 3, 3, 3, 4],

[1, 2, 2, 2, 2, 3, 4], [1, 2, 2, 3, 3, 3, 3], [1, 2, 2, 2, 2, 3, 3],

[1, 2, 2, 4, 4, 4, 4], [1, 2, 2, 2, 2, 4, 4], [1, 3, 3, 4, 4, 4, 4],

[1, 3, 3, 3, 3, 4, 4], [2, 3, 3, 4, 4, 4, 4], [2, 3, 3, 3, 3, 4, 4],

[2, 2, 3, 4, 4, 4, 4], [2, 2, 3, 3, 3, 3, 4], [2, 2, 2, 2, 3, 4, 4],

[2, 2, 2, 2, 3, 3, 4], [2, 2, 2, 3, 3, 3, 3], [2, 2, 2, 2, 3, 3, 3],

[2, 2, 2, 4, 4, 4, 4], [2, 2, 2, 2, 4, 4, 4], [3, 3, 3, 4, 4, 4, 4],

[3, 3, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 3, 3], [1, 2, 4, 4, 4, 4, 4],

[1, 3, 4, 4, 4, 4, 4], [1, 3, 3, 3, 3, 3, 4], [2, 3, 4, 4, 4, 4, 4],

[2, 3, 3, 3, 3, 3, 4], [2, 2, 3, 3, 3, 3, 3], [2, 2, 4, 4, 4, 4, 4],

[3, 3, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 4, 4], [1, 3, 3, 3, 3, 3, 3],

[1, 4, 4, 4, 4, 4, 4], [2, 3, 3, 3, 3, 3, 3], [2, 4, 4, 4, 4, 4, 4],

[3, 4, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 3, 4], [3, 3, 3, 3, 3, 3, 3],

[4, 4, 4, 4, 4, 4, 4].

MAPLE

with(combinat)

kend := 4;

Liste := NULL;

for k from 0 to kend do

Liste := Liste, `$`(k, k^2)

end do;

Liste := [Liste];

for k from 0 to 2^(kend+1)-1 do

Teilergebnis[k] := choose(Liste, k)

end do;

seq(nops(Teilergebnis[k]), k = 0 .. 2^(kend+1)-1)

' Excel VBA

Sub A180174()

Dim n As Long, nend As Long, k As Long, kk As Long, length_row As Long, length_sum As Long

Dim ATable(10, -1000 To 1000) As Double, Summe As Double

Dim offset_row As Integer, offset_column As Integer

Worksheets("Tabelle2").Select

Cells.Select

Selection.ClearContents

Range("A1").Select

offset_row = 1

offset_column = 1

nend = 7

ATable(0, 0) = 1

Cells(0 + offset_row, 0 + offset_column) = 1

For n = 1 To nend

length_row = n * (n + 1) * (2 * n + 1) / 6

length_sum = n ^ 2 + 1

For k = 0 To length_row / 2

Summe = 0

For kk = k - length_sum + 1 To k

Summe = Summe + ATable(n - 1, kk)

Next kk

ATable(n, k) = Summe

Cells(n + offset_row, k + offset_column) = ATable(n, k)

ATable(n, length_row - k) = Summe

Cells(n + offset_row, length_row - k + 0 + offset_column) = ATable(n, k)

Next k

Next n

End Sub

CROSSREFS

Cf. A007318, A008302, A028361, A056520, A000096, A005581, A005582.

Sequence in context: A037812 A037200 A263992 * A172363 A181877 A236472

Adjacent sequences:  A180171 A180172 A180173 * A180175 A180176 A180177

KEYWORD

nonn,tabf

AUTHOR

Thomas Wieder, Aug 15 2010

STATUS

approved

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Last modified October 20 18:39 EDT 2019. Contains 328269 sequences. (Running on oeis4.)