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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 G. C. Greubel, Table of n, a(n) for n = 0..1500 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 Cf. A002571, A203804, A203805, A203806, A203807, A203808, A203809. Cf. A203853, A203800. Sequence in context: A330207 A104317 A178933 * A115664 A182753 A263124 Adjacent sequences: A203800 A203801 A203802 * A203804 A203805 A203806 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.)