A136158 Sequence whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

1, 1, 1, 3, 4, 1, 9, 15, 7, 1, 27, 54, 36, 10, 1, 81, 189, 162, 66, 13, 1, 243, 648, 675, 360, 105, 16, 1, 729, 2187, 2673, 1755, 675, 153, 19, 1, 2187, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1

Offset

0,4

Comments

Row sums = A081294: (1, 2, 8, 32, 128, 512, ...).

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 17 2007

Equals A080419 when first column is removed. - Georg Fischer, Jul 25 2023

Links

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

Formula

Given A136157 = M, an infinite lower triangular bidiagonal matrix with (3, 3, 3, ...) in the main diagonal, (1, 1, 1, ...) in the subdiagonal and the rest zeros; rows of A136157 are generated from M^n * [1, 1, 0, 0, 0, ...], given a(0) = 1.

T(n,k) = A038763(n,n-k). - Philippe Deléham, Dec 17 2007

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) for n > 1, T(0,0) = T(1,1) = T(1,0) = 1. - Philippe Deléham, Oct 30 2013

Sum_{k=0..n} T(n,k)*x^k = (1+x)*(3+x)^(n-1), n >= 1. - Philippe Deléham, Oct 30 2013

G.f.: (-1+2*x)/(-1+3*x+x*y). - R. J. Mathar, Aug 11 2015

Example

First few rows of the triangle:
    1;
    1,    1;
    3,    4,    1;
    9,   15,    7,    1;
   27,   54,   36,   10,   1;
   81,  189,  162,   66,  13,   1;
  243,  648,  675,  360, 105,  16,  1;
  729, 2187, 2673, 1755, 675, 153, 19, 1;
  ...

Prog

(PARI) T(n, k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); 3*T(n-1, k) + T(n-1, k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 25 2023

Crossrefs

Cf. A080419, A081294, A136157.

Sequence in context: A198739 A158076 A010611 * A164948 A102051 A078068

Adjacent sequences: A136155 A136156 A136157 * A136159 A136160 A136161

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 16 2007

Extensions

More terms from Philippe Deléham, Dec 17 2007

Status

approved