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A357537
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a(n) = 2*A080635(n) if n > 0. a(0) = 1.
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0
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1, 2, 2, 6, 18, 78, 378, 2214, 14562, 108702, 897642, 8171766, 81066258, 871695918, 10091490138, 125189658054, 1656458307522, 23288226400062, 346663764078282, 5447099463010326, 90094171024954098, 1564653992673809358, 28467075416816935098, 541467979789775621094
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: sqrt(3) tan(Pi/6 + x sqrt(3)/2).
E.g.f. A(x) satisfies 2*A' = 3 + A^2, A'' = A*A'.
Let f(x) = 1 + x + x^2. Then a(n+1) = (f(x)*d/dx)^n f'(x) evaluated at x = 0.
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EXAMPLE
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G.f. = 1 + 2*x + 2*x^2 + 6*x^3 + 18*x^4 + 78*x^5 + 378*x^6 + ...
E.g.f. = 1 + 2*x + x^2 + x^3 + 3/4*x^4 + 13/20*x^5 + 21/40*x^6 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n! Simplify@SeriesCoefficient[ Sqrt[3] Tan[ Pi/6 + x Sqrt[3]/2], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, Nest[Expand[(1 + x + x^2) D[#, x]]&, 1 + 2 x, n] /. x->0];
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PROG
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(PARI) {a(n) = my(A); if(n<0, 0, A = 1 + 2*x; for( k=1, n, A = A' * (1 + x + x^2)); polcoeff(A, 0))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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