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A193725 Mirror of the triangle A193724. 3
1, 1, 1, 3, 5, 2, 9, 21, 16, 4, 27, 81, 90, 44, 8, 81, 297, 432, 312, 112, 16, 243, 1053, 1890, 1800, 960, 272, 32, 729, 3645, 7776, 9180, 6480, 2736, 640, 64, 2187, 12393, 30618, 43092, 37800, 21168, 7392, 1472, 128, 6561, 41553, 116640, 190512, 199584, 139104, 64512, 19200, 3328, 256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A193725 is obtained by reversing the rows of the triangle A193724.
Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [1,1,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
LINKS
FORMULA
Write w(n,k) for the triangle at A193724. The triangle at A193725 is then given by w(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - Philippe Deléham, Oct 05 2011
G.f.: (-1+2*x+x*y)/(-1+3*x+2*x*y). - R. J. Mathar, Aug 11 2015
EXAMPLE
First six rows:
1;
1, 1;
3, 5, 2;
9, 21, 16, 4;
27, 81, 90, 44, 8;
81, 297, 432, 312, 112, 16;
MATHEMATICA
z = 8; a = 1; b = 2; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193724 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193725 *)
CROSSREFS
Sequence in context: A057033 A003574 A101157 * A077952 A077975 A356378
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)