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%I #7 Dec 16 2014 15:05:23
%S 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,
%T 45,55,65,75,84,91,96,99,100,100,100,99,96,91,84,75,65,55,45,35,25,16,
%U 9,4,1,1,5,14,30,55,90,135,190,255,330,414,505,601,700,800,900,1000,1099
%N Triangle read by rows of the numbers C(n,k) of k-subsets of a quadratically populated n-multiset M.
%C The multiplicity m(i) of the i-th element with 1 <= i <= n is m(i)=i^2.
%C Thus M=[1,2,2,2,2,...,i^2 x i,...,n^2 x n].
%C Row sum is equal to A028361.
%C Column for k=2 is equal to AA000096.
%C Column for k=3 is equal to AA005581.
%C Column for k=4 is equal to AA005582.
%C The number of coefficients C(n,k) for given n is equal to A056520.
%F C(0,0) = 0.
%F C(n,k) = sum_{j=(k-LS+1)}^{k} C(n-1,j).
%F for n > 0 and k=1,...,LR with LS = n^2+1 and LR = n*(n+1)*(2*n+1)/6.
%F C(n,k) = C(n,LR-k).
%e 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].
%e For k=7 we have 55 subsets from M:
%e [1, 2, 2, 3, 3, 4, 4], [1, 2, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 4, 4],
%e [1, 2, 2, 3, 4, 4, 4], [1, 2, 2, 3, 3, 3, 4], [1, 2, 2, 2, 3, 4, 4],
%e [1, 2, 2, 2, 3, 3, 4], [2, 2, 3, 3, 4, 4, 4], [2, 2, 3, 3, 3, 4, 4],
%e [2, 2, 2, 3, 3, 4, 4], [1, 2, 2, 2, 3, 3, 3], [1, 2, 2, 2, 4, 4, 4],
%e [1, 3, 3, 3, 4, 4, 4], [2, 3, 3, 3, 4, 4, 4], [2, 2, 2, 3, 4, 4, 4],
%e [2, 2, 2, 3, 3, 3, 4], [1, 2, 3, 4, 4, 4, 4], [1, 2, 3, 3, 3, 3, 4],
%e [1, 2, 2, 2, 2, 3, 4], [1, 2, 2, 3, 3, 3, 3], [1, 2, 2, 2, 2, 3, 3],
%e [1, 2, 2, 4, 4, 4, 4], [1, 2, 2, 2, 2, 4, 4], [1, 3, 3, 4, 4, 4, 4],
%e [1, 3, 3, 3, 3, 4, 4], [2, 3, 3, 4, 4, 4, 4], [2, 3, 3, 3, 3, 4, 4],
%e [2, 2, 3, 4, 4, 4, 4], [2, 2, 3, 3, 3, 3, 4], [2, 2, 2, 2, 3, 4, 4],
%e [2, 2, 2, 2, 3, 3, 4], [2, 2, 2, 3, 3, 3, 3], [2, 2, 2, 2, 3, 3, 3],
%e [2, 2, 2, 4, 4, 4, 4], [2, 2, 2, 2, 4, 4, 4], [3, 3, 3, 4, 4, 4, 4],
%e [3, 3, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 3, 3], [1, 2, 4, 4, 4, 4, 4],
%e [1, 3, 4, 4, 4, 4, 4], [1, 3, 3, 3, 3, 3, 4], [2, 3, 4, 4, 4, 4, 4],
%e [2, 3, 3, 3, 3, 3, 4], [2, 2, 3, 3, 3, 3, 3], [2, 2, 4, 4, 4, 4, 4],
%e [3, 3, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 4, 4], [1, 3, 3, 3, 3, 3, 3],
%e [1, 4, 4, 4, 4, 4, 4], [2, 3, 3, 3, 3, 3, 3], [2, 4, 4, 4, 4, 4, 4],
%e [3, 4, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 3, 4], [3, 3, 3, 3, 3, 3, 3],
%e [4, 4, 4, 4, 4, 4, 4].
%p with(combinat)
%p kend := 4;
%p Liste := NULL;
%p for k from 0 to kend do
%p Liste := Liste, `$`(k, k^2)
%p end do;
%p Liste := [Liste];
%p for k from 0 to 2^(kend+1)-1 do
%p Teilergebnis[k] := choose(Liste, k)
%p end do;
%p seq(nops(Teilergebnis[k]), k = 0 .. 2^(kend+1)-1)
%p ' Excel VBA
%p Sub A180174()
%p Dim n As Long, nend As Long, k As Long, kk As Long, length_row As Long, length_sum As Long
%p Dim ATable(10, -1000 To 1000) As Double, Summe As Double
%p Dim offset_row As Integer, offset_column As Integer
%p Worksheets("Tabelle2").Select
%p Cells.Select
%p Selection.ClearContents
%p Range("A1").Select
%p offset_row = 1
%p offset_column = 1
%p nend = 7
%p ATable(0, 0) = 1
%p Cells(0 + offset_row, 0 + offset_column) = 1
%p For n = 1 To nend
%p length_row = n * (n + 1) * (2 * n + 1) / 6
%p length_sum = n ^ 2 + 1
%p For k = 0 To length_row / 2
%p Summe = 0
%p For kk = k - length_sum + 1 To k
%p Summe = Summe + ATable(n - 1, kk)
%p Next kk
%p ATable(n, k) = Summe
%p Cells(n + offset_row, k + offset_column) = ATable(n, k)
%p ATable(n, length_row - k) = Summe
%p Cells(n + offset_row, length_row - k + 0 + offset_column) = ATable(n, k)
%p Next k
%p Next n
%p End Sub
%Y Cf. A007318, A008302, A028361, A056520, A000096, A005581, A005582.
%K nonn,tabf
%O 0,5
%A _Thomas Wieder_, Aug 15 2010
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