A111049 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 11, 27, 25, 8, 1, 20, 70, 100, 65, 16, 1, 37, 170, 330, 325, 161, 32, 1, 70, 399, 980, 1295, 966, 385, 64, 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128, 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256

Offset: 0

Author

Philippe Deléham, Oct 07 2005

Keywords

• easy
• nonn
• tabl

Examples

			Rows begin:
  1;
  1,   1;
  1,   3,    2;
  1,   6,    9,    4;
  1,  11,   27,   25,     8;
  1,  20,   70,  100,    65,    16;
  1,  37,  170,  330,   325,   161,    32;
  1,  70,  399,  980,  1295,   966,   385,   64;
  1, 135,  917, 2723,  4515,  4501,  2695,  897,  128;
  1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;
		

Crossrefs

Cf. A002064, A006127.

Programs

Mathematica

With[{m = 9}, CoefficientList[CoefficientList[Series[(1 - 2*x - 2*x*y + x^2 *y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2), {x, 0 , m}, {y, 0, m} ], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

PARI

T(n, k) = if (k<=n, 2^(n-1)*binomial(n-1, k-1)+binomial(n-1, k));
matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020

Formula

T(n, k) = 2^(n-1)binomial(n-1, k-1) + binomial(n-1, k).

Sum_{k=0..n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n >= 1.

From Peter Bala, Mar 20 2013: (Start)

O.g.f.: (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....

E.g.f.: (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....

Recurrence equation: for n >= 1, T(n+1,k) = 2*T(n,k) + 2*T(n,k-1) - binomial(n,k). (End)

From Philippe Deléham, Oct 18 2013: (Start)

G.f.: (1 - 2*x - 2*x*y + x^2*y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2).

T(n,k) = 3*T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k) - 4*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = T(1,1) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(2,2) = 2, T(n,k) = 0 if k > n or if k < 0. (End)

Extensions

Wrong a(42) removed by Georg Fischer, Feb 17 2020