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%I #8 Oct 04 2022 07:29:41
%S 1,2,2,6,18,78,378,2214,14562,108702,897642,8171766,81066258,
%T 871695918,10091490138,125189658054,1656458307522,23288226400062,
%U 346663764078282,5447099463010326,90094171024954098,1564653992673809358,28467075416816935098,541467979789775621094
%N a(n) = 2*A080635(n) if n > 0. a(0) = 1.
%F E.g.f.: sqrt(3) tan(Pi/6 + x sqrt(3)/2).
%F E.g.f. A(x) satisfies 2*A' = 3 + A^2, A'' = A*A'.
%F Let f(x) = 1 + x + x^2. Then a(n+1) = (f(x)*d/dx)^n f'(x) evaluated at x = 0.
%e G.f. = 1 + 2*x + 2*x^2 + 6*x^3 + 18*x^4 + 78*x^5 + 378*x^6 + ...
%e E.g.f. = 1 + 2*x + x^2 + x^3 + 3/4*x^4 + 13/20*x^5 + 21/40*x^6 + ...
%t a[ n_] := If[ n < 0, 0, n! Simplify@SeriesCoefficient[ Sqrt[3] Tan[ Pi/6 + x Sqrt[3]/2], {x, 0, n}]];
%t a[ n_] := If[ n < 0, 0, Nest[Expand[(1 + x + x^2) D[#, x]]&, 1 + 2 x, n] /. x->0];
%o (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))};
%Y Cf. A080635.
%K nonn
%O 0,2
%A _Michael Somos_, Oct 02 2022
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