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

0,5

Links

Table of n, a(n) for n=0..54.

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A002064, A006127.

Sequence in context: A181897 A337977 A212207 * A211955 A088617 A190909

Adjacent sequences: A111046 A111047 A111048 * A111050 A111051 A111052

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Oct 07 2005

Extensions

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

Status

approved