A123191 -id:A123191 - cp's OEIS frontend

A156710 Triangle read by rows, A123191 * (A002605 * (A002605 * 0^(n-k))).

1, 1, 1, 1, 3, 1, 1, 3, 6, 6, 1, 3, 6, 18, 16, 1, 3, 6, 18, 48, 44, 1, 3, 6, 18, 48, 132, 120, 136, 18, 48, 132, 360, 328, 1, 3, 6, 18, 48, 132, 360, 984, 896, 1, 3, 6, 18, 48, 132, 360, 984, 2688, 2448, 1, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 6688

Offset: 0

Gary W. Adamson, Feb 14 2009

Row sums = A002605: (1, 2, 6, 16, 44, 120,...)

As a property of eigentriangles, sum of row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 3, 6, 6;
1, 3, 6, 18, 16;
1, 3, 6, 18, 48, 44;
1, 3, 6, 18, 48, 132, 120;
1, 3, 6, 18, 48, 132, 360, 328;
1, 3, 6, 18, 48, 132, 360, 984, 896;
1, 3, 6, 18, 48, 132, 360, 984, 2688, 2448;
1, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 6688;
...

Cf. A002605, A123191.

Triangle read by rows, A123191 * (A002605 * (A002605 * 0^(n-k))). A123191 is
unsigned, (A002605 * 0^(n-k))= an infinite lower triangular matrix with
A002605 as the main diagonal prefaced with a 1: (1, 1, 2, 6, 16, 44,...)
and the rest zeros.

A193559 Augmentation of the triangular array |A123191|. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 7, 17, 7, 1, 10, 41, 82, 32, 1, 13, 74, 238, 434, 166, 1, 16, 116, 502, 1412, 2446, 926, 1, 19, 167, 901, 3317, 8587, 14405, 5419, 1, 22, 227, 1462, 6581, 21802, 53381, 87610, 32816, 1, 25, 296, 2212, 11717, 46681, 143666, 338038

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

			First five rows of |A123191|:
1
1...1
1...3...1
1...3...3...1
1...3...3...3...1
First 5 rows of A193559:
1
1...1
1...4...2
1...7...17...7
1...10..41...82...32

Cf. A193091.

p[n_, k_] := If[Or[k == 0, k == n], 1, 3]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* Abs. value of A123191 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193559 *)
Flatten[Table[v[n], {n, 0, 10}]]

cp's OEIS Frontend

A156710 Triangle read by rows, A123191 * (A002605 * (A002605 * 0^(n-k))).

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