A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Author

Philippe Deléham, Dec 14 2011

Keywords

• nonn
• tabl

Comments

A071919*A007318 as infinite lower triangular matrices.

A129186*A038207 as infinite lower triangular matrices.

From Paul Curtz, Nov 12 2019: (Start)

If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:

The first column is A166444.

The second column is A139756.

The antidiagonal sums are A000129 (Pell numbers).

The row sums are (-1)^n*A141413.

The signed row sums are 0 followed by 1's, autosequence companion to A054977.

(End)

Examples

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...
		

Crossrefs

Essentially the same as A038207, A062715, A065109.

Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).

Cf. A141413, A166444, A054977.

Programs

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

Mathematica

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .

G.f.: (1-(1+y)*x)/(1-(2+y)*x).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.

T(2n,n) = A069720(n).