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A006673
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E.g.f. is the logarithmic derivative of e.g.f. for Pell numbers [1, 0, 1, 2, 5, ...].
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1
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0, 1, 2, 2, -8, -56, -112, 848, 9088, 25216, -310528, -4334848, -14701568, 270029824, 4554426368, 17536821248, -458243735552, -8926669144064, -37024075153408, 1341521605885952, 29290212127670272, 125297096967061504, -6224109737622372352
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OFFSET
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0,3
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COMMENTS
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It looks like the signs have a repeating pattern of +, +, +, -, -, -, etc.; however, a(42) breaks this pattern by being positive after a string of only two negative terms. The sequence oscillates because it has two dominant singularities; the oscillation is irregular and unpredictable because the arguments of the singularities are not rational multiples of Pi; the oscillation first appears regular with period 6 because the arguments of the singularities are +- 1.01*Pi/3, which is very close to 1/6 of the circle. All of this can be proved with the standard techniques of analytic combinatorics (see the Flajolet and Sedgewick reference). - Justin M. Troyka, Jun 20 2019
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REFERENCES
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P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, Cambridge, 2009, pages 258-259 ("Expansion of meromorphic functions") and 264-266 ("Nonperiodic fluctuations").
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LINKS
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FORMULA
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G.f.: 1-2/Q(0) where Q(k) = 1 + 1/(1 + 2*(k+1)/(-1/x +(2*k+2)/Q(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 23 2012
G.f.: -1/Q(0), where Q(k) = 2*k+2 - 1/x + (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: 1/Q(0), where Q(k)= 1/(x*(k+1)) - 2 + 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
G.f.: x/Q(0), m=-2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
E.g.f.: 2*x/Q(0), where Q(k) = 8*k+2 - 2*x/(1 + 2*x/(4*k+3 - 2*x/(1 + 2*x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2013
E.g.f.: sqrt(2)/(sqrt(2) - tanh(sqrt(2)*x)) -1 = W(0) -1, where W(k) = 1 + x/(4*k+1 - 1*x/(1 + 2*x/(4*k+3 - 2*x/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2014
a(n) ~ (-2) cos((n+1)*t) R^(-(n+1)) n!, where t = arctan(y/x) and R = sqrt(x^2+y^2), where x = log(3+2*sqrt(2))/(2*sqrt(2)) and y = Pi/(2*sqrt(2)) (note that R is approximately 1.274 and t is approximately 1.012*Pi/3). This formula follows from the standard techniques of analytic combinatorics (see the Flajolet and Sedgewick reference). - Justin M. Troyka, Jun 20 2019
a(n+1) = 2*a(n) - Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k) if n>=1. - Michael Somos, Apr 22 2020
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EXAMPLE
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G.f. = x + 2*x^2 + 2*x^3 - 8*x^4 - 56*x^5 - 112*x^6 + 848*x^7 + 9088*x^8 + ...
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MAPLE
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# After Sergei N. Gladkovskii.
seq(k!*coeff(series(1/(sqrt(2)*coth(sqrt(2)*x)-1), x=0, k+2), x, k), k=0..21); # Peter Luschny, Nov 18 2014
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (1 - Tanh[ x Sqrt[2]] / Sqrt[2]) - 1, {x, 0, n}]]; (* Michael Somos, Nov 22 2014 *)
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PROG
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(PARI) {a(n) = my(w=quadgen(8)); if( n<0, 0, n! * polcoeff( 1 / (1 - tanh(w*x + x * O(x^n)) / w) - 1, n))}; /* Michael Somos, Nov 22 2014 */
(PARI) {a(n) = n--; if(n < 1, n==0, 2*a(n) - sum(k=1, n-1, binomial(n, k) * a(k) * a(n-k)))}; /* Michael Somos, Apr 22 2020 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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