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A206140
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G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - Lucas(k)*x^k + (-1)^k*x^(2*k)).
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0
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1, 1, 2, 4, 10, 18, 44, 78, 178, 329, 699, 1299, 2724, 5013, 10198, 19076, 37786, 70448, 138322, 256649, 497884, 925309, 1770195, 3283971, 6249417, 11542969, 21785791, 40231877, 75388314, 138806128, 259010824, 475364152, 882643035, 1617443421, 2988026528
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OFFSET
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0,3
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COMMENTS
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Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).
As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - u^k*x - (u^k*x)^2).
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LINKS
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Table of n, a(n) for n=0..34.
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 18*x^5 + 44*x^6 + 78*x^7 +...
where
A(x) = 1 + x/(1-x-x^2) + x^2/((1-x-x^2)*(1-3*x^2+x^4)) + x^3/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)) + x^4/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)*(1-7*x^4+x^8)) + x^5/((1-x-x^2)*(1-3*x^2+x^4)*(1-4*x^3-x^6)*(1-7*x^4+x^8)*(1-11*x^5-x^10)) +...).
The Lucas numbers begin:
A000204 = [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ...].
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PROG
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(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=0, n, x^m/prod(k=1, m, 1-Lucas(k)*x^k+(-1)^k*x^(2*k)+x*O(x^n))), n)}
for(n=0, 51, print1(a(n), ", "))
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CROSSREFS
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Cf. A000204, A206141.
Sequence in context: A303438 A348396 A104723 * A079162 A257593 A197926
Adjacent sequences: A206137 A206138 A206139 * A206141 A206142 A206143
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Feb 04 2012
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STATUS
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approved
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