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A147542
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Product(1 + a(n)*x^n, n=1..infinity) = sum(F(k+1)*x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).
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8
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1, 2, 1, 4, 2, 1, 4, 18, 8, 8, 18, 17, 40, 50, 88, 396, 210, 296, 492, 690, 1144, 1776, 2786, 3545, 6704, 10610, 16096, 25524, 39650, 63544, 97108, 269154, 236880, 389400, 589298, 956000, 1459960, 2393538, 3604880, 5739132, 9030450, 14777200
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A formal infinite product representation for the Fibonacci numbers (A000045(n+1)).
For references see A147541. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009]
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LINKS
| W. Lang Two recurrences for the general problem.
R. J. Mathar, Re: polynomial-to-product transform, Maple code (2008). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009]
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FORMULA
| Comments from Wolfdieter Lang, Mar 06 2009 (Start): Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)):
a(n)= F(n+1) - sum(sum(product(a(k[j]),j=1..m),fp from FP(n,m)),m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=F(2)=1, a(2)=F(3)=2. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With the definition of FP(n,m) from the above recurrence I, P(n,m) the general set of partitions of n with m parts, and the multinomial numbers M_0 (given for every partition under A048996):
a(n) = sum((d/n)*(-a(d)^(n/d)),d|n with 1<d<n) + sum(((-1)^(m-1))*(1/m)*sum(M_0(p)*F(2)^e(1)*...*F(n+1)^e(n),p=(1^e(1),...,n^e(n)) from P(n,m)),m=1..n-1), n>=2; a(1)=F(2)=1. The exponents e(j)>=0 satisfy sum(j*e(j),j=1..n)=n and sum(e(j),j=1..m). The M_0 numbers are m!/product(e(j)!,j=1..n).
Example of recurrence I: a(4) = F(5) - a(1)*a(3) = 5 - 1*1 = 4.
Example of recurrence II: a(4)= 2*(-1)^2 + (1*F(5)-(1/2)*(2*F(2)*F(4) + 1*F(3)^2) + (1/3)*3*F(2)^2*F(3)) = 4. (End)
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CROSSREFS
| Cf. A000045, A137852, A006973, A157159.
Sequence in context: A118235 A083653 A135152 * A128177 A087738 A133113
Adjacent sequences: A147539 A147540 A147541 * A147543 A147544 A147545
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KEYWORD
| nonn
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AUTHOR
| N. Fernandez (primeness(AT)borve.org), Nov 06 2008
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EXTENSIONS
| More terms and revised description from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 06 2009
Edited by N. J. A. Sloane, Mar 11 2009 at the suggestion of Vladeta Jovovic
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009
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