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A203803 G.f.: exp( Sum_{n>=1} A000204(n)^3 * x^n/n ) where A000204 is the Lucas numbers. 11
1, 1, 14, 35, 205, 744, 3414, 13926, 60060, 252330, 1072902, 4537272, 19234463, 81452015, 345084970, 1461714517, 6192083147, 26229794928, 111111714300, 470675847900, 1993816532280, 8445939457380, 35777578796220, 151556246864400, 642002579853325, 2719566542567917 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).
LINKS
FORMULA
G.f.: 1/( (1+x-x^2)^3 * (1-4*x-x^2) ).
G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203853(n) where A203853(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^2.
EXAMPLE
G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 +...
where
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 +...+ Lucas(n)^3*x^n/n +...
MATHEMATICA
CoefficientList[Series[1/((1 + x - x^2)^3*(1 - 4*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 24 2017 *)
PROG
(PARI) /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^3*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}
CROSSREFS
Sequence in context: A330207 A104317 A178933 * A115664 A182753 A263124
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 06 2012
STATUS
approved

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Last modified April 14 03:47 EDT 2024. Contains 371652 sequences. (Running on oeis4.)