OFFSET
0,2
FORMULA
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.
EXAMPLE
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 + ...
MATHEMATICA
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];
PROG
(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))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 02 2022
STATUS
approved