A262964 Lower triangular matrix. See comments for definition.

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 2, 1, 0, 10, 4, 9, 0, 1, 0, 15, 10, 28, 7, 5, 1, 0, 21, 20, 69, 36, 30, 1

Offset: 1

Author

Mats Granvik, Oct 05 2015

Keywords

• tabl
• nonn

Comments

Handmade table created by looking at the "columns" that are separated by zeros in the Mathematica program output. First column is A000012. Second column A000004. Third column A000217. Fourth column A000292. At least a few columns onwards are recognized by the OEIS automated sequence recognition as polynomials.

Row sums are A239605. Alternating row sums are A011782.

Examples

			Triangle starts and row sums are A239605:
   1                                    =    1
   1    0                               =    1
   1    0    1                          =    2
   1    0    3    0                     =    4
   1    0    6    1    2                =   10
   1    0   10    4    9    0           =   24
   1    0   15   10   28    7    5      =   66
   1    0   21   20   69   36   30    1 =  178
Alternating row sums are A011782:
   1                                    =    1
   1   -0                               =    1
   1   -0    1                          =    2
   1   -0    3   -0                     =    4
   1   -0    6   -1    2                =    8
   1   -0   10   -4    9   -0           =   16
   1   -0   15  -10   28   -7    5      =   32
   1   -0   21  -20   69  -36   30   -1 =   64
		

Crossrefs

Cf. Row sums A239605, alternating row sums A011782, columns: A000012, A000004, A000217, A000292.

Programs

Mathematica

(* To get this number triangle, look at the numbers separated by zeros in the columns *)
(* coefficients (coeff) in power series can be changed *)Clear[t, n, k, i, nn, x];
Clear[x]
coeff = {1, 1000000000000000000, 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, 1, 1};
mp[m_, e_] :=
If[e == 0, IdentityMatrix@Length@m, MatrixPower[m, e]]; nn =
Length[coeff]; cc = Range[nn]*0 + 1; Monitor[
Do[Clear[t]; t[n_, 1] := t[n, 1] = cc[[n]];
  t[n_, k_] :=
   t[n, k] =
    If[n >= k,
     Sum[t[n - i, k - 1], {i, 1, k - 1}] -
      Sum[t[n - i, k], {i, 1, k - 1}], 0];
  A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  A5 = A4[[1 ;; nn - 1]]; A5 = Prepend[A5, ConstantArray[0, nn]];
  cc = Total[
    Table[coeff[[n]]*mp[A5, n - 1][[All, 1]], {n, 1, nn}]];, {i, 1,
   nn}], i];
cc;
TableForm[A4[[All,1]]]