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Search: id:A118184
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| A118184 |
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Column 0 of the matrix log of triangle A118180, after term in row n is multiplied by n: a(n) = n*[LOG(A118180)](n,0), where A118180(n,k) = (3^k)^(n-k). |
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+0
4
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| 0, 1, -1, 3, -23, 329, 18231, -22030373, 34718491601, -130548608723439, 1300095260497408879, -35497483240662990289357, 2687397326811421691366217657, -562747611676887059779727492799911, 320110532506391993959111359699070808231
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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The entire matrix log of triangle A118180 is determined by column 0 (this sequence): [LOG(A118180)](n,k) = a(n-k)/(n-k)*(3^k)^(n-k) for n>k>=0.
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FORMULA
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G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-3^n*x). By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118183(n-k)*(3^k)^(n-k) for n>=0. a(3^n) is divisible by 3^n.
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EXAMPLE
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Column 0 of LOG(A118180) = [0, 1, -1/2, 3/3, -23/4, 329/5, 18231/6,...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...
= x/(1-3*x) - x^2/(1-9*x) + 3*x^3/(1-27*x) - 23*x^4/(1-81*x) + 329*x^5/(1-243*x) + 18231*x^6/
(1-729*x) - 22030373*x^7/(1-2187*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, (3^(c-1))^(r-c))), L=sum(m=1, #T, -(T^0-T)^m/m)); return(n*L[n+1, 1])}
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CROSSREFS
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Cf. A118180, A118183.
Sequence in context: A052842 A088692 A129458 this_sequence A027486 A092664 A073588
Adjacent sequences: A118181 A118182 A118183 this_sequence A118185 A118186 A118187
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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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