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

0,4

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)

Links

Table of n, a(n) for n=0..65.

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).

Example

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;
  ...

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 *)

Crossrefs

Essentially the same as A038207, A062715, A065109.

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

Cf. A141413, A166444, A054977.

Sequence in context: A218599 A051623 A244124 * A296129 A276544 A214753

Adjacent sequences: A183187 A183188 A183189 * A183191 A183192 A183193

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 14 2011

Status

approved