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A227009 Irregular triangle read by rows: T(n,k) is the number of partitions of an n X n square lattice into squares that contain k nodes unconnected to any of their neighbors, considering only the number of parts. 2
1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 4, 4, 4, 3, 4, 3, 2, 2, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

The n-th row contains (n-1)^2 + 1 elements.

The irregular triangle is shown below.

\ k 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...

n

1   1

2   1  1

3   1  1  0  0  1

4   1  1  1  1  2  0  0  0  0  1

5   1  1  1  1  2  1  1  1  0  1  0  0  0  0  0  0  1

6   1  1  1  1  2  2  2  2  3  4  2  2  2  2  1  0  2  0  0 ...

7   1  1  1  1  2  2  2  2  3  4  3  3  4  4  4  3  4  3  2 ...

LINKS

Christopher Hunt Gribble and Alois P. Heinz, Rows n = 1..13, flattened (Rows n = 1..7 from Christopher Hunt Gribble)

FORMULA

It appears that T(n,k) = T(n-1,k), n odd, n > 1 and k = 0..(n-1)^2/4.

Sum_{k=0..(n-1)^2} T(n,k) = A034295(n).

EXAMPLE

For n = 6, there are 3 partitions that contain 8 isolated nodes, so T(6,8) = 3.

An m X m square contains (m-1)^2 isolated nodes.

Consider that each partition is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  Then the 3 partitions are:

1 1 1 1 1 1 1    1 1 1 1 1 1 1    1 1 1 1 1 1 1

1 0 1 0 1 0 1    1 0 0 1 1 0 1    1 0 0 1 0 0 1

1 1 1 1 1 1 1    1 0 0 1 1 1 1    1 0 0 1 0 0 1

1 0 1 0 1 0 1    1 1 1 1 1 0 1    1 1 1 1 1 1 1

1 1 1 1 1 1 1    1 1 1 1 1 1 1    1 1 1 1 1 1 1

1 0 1 0 1 1 1    1 1 1 0 1 0 1    1 1 1 1 1 1 1

1 1 1 1 1 1 1    1 1 1 1 1 1 1    1 1 1 1 1 1 1

MAPLE

b:= proc(n, l) option remember; local i, k, s, t;

      if max(l[])>n then {} elif n=0 or l=[] then {0}

    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))

    else for k do if l[k]=0 then break fi od; s:={};

         for i from k to nops(l) while l[i]=0 do s:=s union

             map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,

                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))

         od; s

      fi

    end:

T:= n-> (w->seq(coeff(w, z, h), h=0..(n-1)^2))(add(z^add(

    coeff(p, x, i)*(i-1)^2, i=2..degree(p)), p=b(n, [0$n]))):

seq(T(n), n=1..9);  # Alois P. Heinz, Jun 27 2013

MATHEMATICA

b[n_, l_List] := b[n, l] = Module[{i, k , s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {0}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, For[k = 1, k <= Length[l], k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[# + x^(1+i-k)&, b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; Length[l]]]]]]]; s]]; T[n_] := Function[w, Table[Coefficient[w, z, h], {h, 0, (n-1)^2}]][Sum[ z^Sum[Coefficient[p, x, i]*(i-1)^2, {i, 2, Exponent[p, x]}], {p, b[n, Array[0&, n]]}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-Fran├žois Alcover, Jan 24 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A034295.

Sequence in context: A101257 A321892 A225542 * A144629 A271719 A025907

Adjacent sequences:  A227006 A227007 A227008 * A227010 A227011 A227012

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Jun 27 2013

STATUS

approved

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Last modified April 21 10:03 EDT 2019. Contains 322328 sequences. (Running on oeis4.)