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A116603
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Coefficients in asymptotic expansion of sequence A052129.
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7
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1, 2, -1, 4, -21, 138, -1091, 10088, -106918, 1279220, -17070418, 251560472, -4059954946, 71250808916, -1351381762990, 27552372478592, -601021307680207, 13969016314470386, -344653640328891233, 8997206549370634644
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007) 292-314.
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LINKS
| Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant
Eric Weisstein's World of Mathematics, Goebels Sequence
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant
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FORMULA
| G.f. A(x) satisfies (1+x)^2=A(x)^2/A(x/(1+x)).
A003504(n) ~ C^(2^n)(n+2-1/n+4/n^2-21/n^3+138/n^4-1091/n^5+...) where C=1.04783144757... (see A115632).
A052129(n) ~ s^(2^n)/(n+2-1/n+4/n^2-21/n^3+138/n^4-1091/n^5+...) where s=1.661687949633... (see A112302).
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MATHEMATICA
| n = 19; f[x_] := Sum[c[k] x^k, {k, 0, n}]; c[0] = 1; eq = Thread[ CoefficientList[ Series[(1 + x)^2*f[x/(1 + x)] - f[x]^2, {x, 0, n}], x] == 0]; Table[c[k], {k, 0, n}] /. Solve[ Rest[eq]][[1]] (* From Jean-François Alcover, Jul 28 2011, after g.f. *)
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=1; for(k=1, n, A=truncate(A+O(x^k))+x*O(x^k); A=-A+2/subst(A^(-1/2), x, x/(1+x))*(1+x); ); polcoeff(A, n))}
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CROSSREFS
| Cf. A052129, A112302, A123851, A123852, A123853, A123854.
Sequence in context: A010252 A032105 A053565 * A158356 A015939 A174941
Adjacent sequences: A116600 A116601 A116602 * A116604 A116605 A116606
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Feb 18 2006
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