A128134 A128132 * A007318.

1, 1, 2, 2, 5, 3, 3, 10, 11, 4, 4, 17, 27, 19, 5, 5, 26, 54, 56, 29, 6, 6, 37, 95, 130, 100, 41, 7, 7, 50, 153, 260, 265, 162, 55, 8, 8, 65, 231, 469, 595, 483, 245, 71, 9, 9, 82, 332, 784, 1190, 1204, 812, 352, 89, 10

Offset: 1

Author

Gary W. Adamson, Feb 15 2007

Keywords

• nonn
• tabl

Comments

A007318 * A128132 = A128133. Row sums = A128135: (1, 3, 10, 28, 72, 176, ...).

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
  1;
  1,  2;
  2,  5,  3;
  3, 10, 11,   4;
  4, 17, 27,  19,   5;
  5, 26, 54,  56,  29,  6;
  6, 37, 95, 130, 100, 41, 7;
  ...
		

Crossrefs

Cf. A002522, A007318, A028387, A128132, A128133, A128135, A164845, A332697.

Formula

A128132 * A007318 as infinite lower triangular matrices (assuming the top of the Pascal triangle A007318 is shifted from (0,0) to (1,1)).

From Petros Hadjicostas, Jul 26 2020: (Start)

T(n,k) = n*binomial(n-1, k-1) - binomial(n-2, k-1)*[n <> k] for 1 <= k <= n, where [ ] is the Iverson bracket.

Bivariate o.g.f.: x*y*(1 - x + x^2*(1 + y))/(1 - x*(1 + y))^2.

T(n,k) = T(n-1,k) + T(n-1,k-1) + binomial(n-1,k-1) for 2 <= k <= n with (n,k) <> (2,2).

T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-2,k-2) + 2*T(n-1,k-1) - 2*T(n-2,k-1) for 3 <= k <= n with (n,k) <> (3,3).

T(n,1) = n - 1 for n >= 2.

T(n,2) = A002522(n-1) for n >= 2.

T(n,3) = A164845(n-3) for n >= 3.

T(n,4) = A332697(n-3) for n >= 4.

T(n,n) = n for n >= 1.

T(n,n-1) = A028387(n-2) for n >= 2. (End)