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Search: id:A118188
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| A118188 |
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Column 0 of the matrix inverse of triangle A118185(n,k) = (4^k)^(n-k). |
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+0
3
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| 1, -1, 3, -33, 1407, -237057, 158992383, -425715556353, 4556004503093247, -194971932801554579457, 33370662957719457037287423, -22845215336421444625717664940033, 62557106610069521429900219032249827327, -685195337175488637158242110253091749621661697
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The entire matrix inverse of triangle A118185 is determined by column 0 (this sequence): [A118185^-1](n,k) = a(n-k)*(4^k)^(n-k) for n>=k>=0. Any g.f. of the form: Sum_{k>=0} b(k)*x^k may be expressed as: Sum_{n>=0} c(n)*x^n/(1-4^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*(4^k)^(n-k).
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FORMULA
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G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-4^n*x). 0^n = Sum_{k=0..n} a(k)*(4^k)^(n-k) for n>=0.
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EXAMPLE
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Recurrence at n=4:
0 = a(0)*(4^0)^4 +a(1)*(4^1)^3 +a(2)*(4^2)^2 +a(3)*(4^3)^1 +a(4)*(4^4)^0
= 1*(4^0) - 1*(4^3) + 3*(4^4) - 33*(4^3) + 1407*(4^0).
The g.f. is illustrated by:
1 = 1/(1-x) - 1*x/(1-4*x) + 3*x^2/(1-16*x) - 33*x^3/(1-64*x) +
1407*x^4/(1-256*x) - 237057*x^5/(1-1024*x) + 158992383*x^6/(1-4096*x) +...
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PROGRAM
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(PARI) {a(n)=local(T=matrix(n+1, n+1, r, c, if(r>=c, (4^(c-1))^(r-c)))); return((T^-1)[n+1, 1])}
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CROSSREFS
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Cf. A118185 (triangle).
Sequence in context: A086894 A012487 A113111 this_sequence A126675 A038694 A134477
Adjacent sequences: A118185 A118186 A118187 this_sequence A118189 A118190 A118191
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2006
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