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Search: id:A118180
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| A118180 |
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Triangle T, read by rows, defined by: T(n,k) = (3^k)^(n-k) for n>=k>=0. For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-3^n*x). |
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11
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| 1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 27, 81, 27, 1, 1, 81, 729, 729, 81, 1, 1, 243, 6561, 19683, 6561, 243, 1, 1, 729, 59049, 531441, 531441, 59049, 729, 1, 1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1, 1, 6561, 4782969, 387420489
(list; table; graph; listen)
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OFFSET
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0,5
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FORMULA
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G.f.: A(x,y) = Sum_{n>=0} x^n/(1-3^n*x*y). G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,3*y).
Equals ConvOffsStoT transform of the 3^n series: (1, 3, 9, 27,...); e.g., ConvOffs transform of (1, 3, 9, 27) = (1, 27, 81, 27, 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 21 2008
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EXAMPLE
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A(x,y) = 1/(1-xy) + x/(1-3xy) + x^2/(1-9xy) + x^3/(1-27xy) + ...
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 9, 9, 1;
1, 27, 81, 27, 1;
1, 81, 729, 729, 81, 1;
1, 243, 6561, 19683, 6561, 243, 1;
1, 729, 59049, 531441, 531441, 59049, 729, 1;
1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1; ...
The matrix inverse T^-1 starts:
1;
-1, 1;
2, -3, 1;
-10, 18, -9, 1;
134, -270, 162,-27, 1;
-4942, 10854, -7290, 1458, -81, 1; ...
where [T^-1](n,k) = A118183(n-k)*(3^k)^(n-k).
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MAPLE
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T(n, k)=if(n<k|k<0, 0, (3^k)^(n-k) )
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CROSSREFS
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Cf. A118181 (row sums), A118182 (antidiagonal sums); A118183, A118184; variants: A117401 (q=2), A118185 (q=4), A118190 (q=5).
Cf. A117401 = ConvOffsStoT transform of 2^n.
Sequence in context: A157179 A152655 A144493 this_sequence A045912 A158695 A106268
Adjacent sequences: A118177 A118178 A118179 this_sequence A118181 A118182 A118183
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2006
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