[seqfan] Re: A123712 and A178212

Mats Granvik mgranvik at abo.fi
Fri Feb 10 15:33:27 CET 2012


The table:
https://oeis.org/A123706
mentioned in the previous messages:
http://list.seqfan.eu/pipermail/seqfan/2012-February/016392.html

can be generated with the Mathematica 8 program:

(*start*)
Clear[t];
t[n_, 1] = n;
t[n_, k_] :=
  t[n, k] =
   If[n >= k, Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}],
    0]; A = Table[Table[t[n, k], {k, 12}], {n, 12}];
MatrixForm[A]
MatrixForm[Inverse[A]]
(*end*)

A variant with rows (from the matrix inverse of the Pascal triangle)
as input, gives the Mobius tranform of the corresponding column
in the Pascal triangle as output:


Example 1:
Input:                                   Output:
1,-1,0,0,0,0,0,0...  ->.Matrixinverse -> 1,1,2,2,4,2,6,4,6,4,10,4...
https://oeis.org/A000010

(*start*)
Clear[t];
t[1, 1] = 1;
t[2, 1] = -1;
t[n_, 1] = 0;
t[n_, k_] :=
  t[n, k] =
   If[n >= k, Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}],
    0]; A = Table[Table[t[n, k], {k, 12}], {n, 12}];
MatrixForm[A]
MatrixForm[Inverse[A]]
(*end*)


Example 2:

(*start*)
Clear[t];
t[1, 1] = 1;
t[2, 1] = -2;
t[3, 1] = 1;
t[n_, 1] = 0;
t[n_, k_] :=
  t[n, k] =
   If[n >= k, Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}],
    0]; A = Table[Table[t[n, k], {k, 12}], {n, 12}];
MatrixForm[A]
MatrixForm[Inverse[A]]
(*end*)

Input:                                     Output:
1,-2,1,0,0,0,0,0,0.... ->.Matrixinverse -> 1,2,5,7,14,13,27,26,39,38,65,50...
https://oeis.org/A007438

Mats





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