

A136158


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


6



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
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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


LINKS



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) = 3*T(n1,k) + T(n1,k1) 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)^(n1), n >= 1.  Philippe Deléham, Oct 30 2013


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(n1, k) + T(n1, k1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 25 2023


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EXTENSIONS



STATUS

approved



