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Search: id:A118185
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| A118185 |
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Triangle T, read by rows, defined by: T(n,k) = (4^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/(1-4^n*x). |
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7
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| 1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 64, 256, 64, 1, 1, 256, 4096, 4096, 256, 1, 1, 1024, 65536, 262144, 65536, 1024, 1, 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1, 1, 16384, 16777216, 1073741824, 4294967296, 1073741824, 16777216, 16384, 1
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m, and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118188(n-k)*T(n,k); also, the matrix log is given by [LOG(T)](n,k) = A118189(n-k)*T(n,k).
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FORMULA
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G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y). G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
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EXAMPLE
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A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
Triangle begins:
1;
1, 1;
1, 4, 1;
1, 16, 16, 1;
1, 64, 256, 64, 1;
1, 256, 4096, 4096, 256, 1;
1, 1024, 65536, 262144, 65536, 1024, 1;
1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
The matrix inverse T^-1 starts:
1;
-1, 1;
3, -4, 1;
-33, 48, -16, 1;
1407, -2112, 768, -64, 1;
-237057, 360192, -135168, 12288, -256, 1; ...
where [T^-1](n,k) = A118188(n-k)*(4^k)^(n-k).
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<0, 0, (4^k)^(n-k) )
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CROSSREFS
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Cf. A118186 (row sums), A118187 (antidiagonal sums); A118188, A118189; variants: A117401 (q=2), A118180 (q=3), A118190 (q=5).
Sequence in context: A116469 A010320 A008304 this_sequence A034802 A139167 A015113
Adjacent sequences: A118182 A118183 A118184 this_sequence A118186 A118187 A118188
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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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