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%I #39 May 28 2022 08:06:51
%S 1,6,-18,108,-810,6804,-61236,577368,-5629338,56293380,-574192476,
%T 5950722024,-62482581252,663276631752,-7106535340200,76750581674160,
%U -834662575706490,9132190534200420,-100454095876204620,1110282112315945800
%N Expansion of sqrt(1 + 12*x).
%C This is also the expansion of sqrt(3)*(2*B2inv(x) - 1), where B2inv is the compositional inverse of the Bernoulli polynomial B(2, x) = 1/6 - x + x^2 = (x - 1/2)^2 - 1/12, for x >= 1/2. (see A196838 and A196839 for the Bernoulli polynomials). - _Wolfdieter Lang_, Aug 26 2015
%H Robert Israel, <a href="/A108735/b108735.txt">Table of n, a(n) for n = 0..838</a>
%F From _Wolfdieter Lang_, Aug 26 2015: (Start)
%F G.f.: sqrt(1 + 12*x) = 1 + 6*x*c(-3*x), with the g.f. c of the Catalan numbers A000108.
%F a(n) = -2*(-3)^n*C(n-1), n >= 1, and a(0) = 1, with C(n) = A000108(n). (End)
%F From _Robert Israel_, Aug 27 2015: (Start)
%F D-finite with recurrence: a(n) = (18/n - 12)*a(n-1).
%F a(n) ~ (-1)^(n+1)*12^n/(2*sqrt(Pi)*n^(3/2)). (End)
%F 0 = a(n)*(144*a(n+1) +30*a(n+2)) +a(n+1)*(-6*a(n+1) +a(n+2)) for all n in Z. - _Michael Somos_, Aug 27 2015
%F a(n) = 2*(-1)^(n+1)*A025226(n). - _R. J. Mathar_, Jan 23 2020
%F From _Amiram Eldar_, May 28 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (192 - 36*arcsinh(1/(2*sqrt(3)))/sqrt(13))/169.
%F Sum_{n>=0} (-1)^n/a(n) = (96 - 36*arcsin(1/(2*sqrt(3)))/sqrt(11))/121.
%e G.f. = 1 + 6*x - 18*x^2 + 108*x^3 - 810*x^4 + 6804*x^5 - 61236*x^6 + ...
%p f:= proc(n) option remember; (18/n - 12)*procname(n-1) end proc: f(0):= 1:
%p map(f, [$0..100]); # _Robert Israel_, Aug 27 2015
%t CoefficientList[Series[(1 + 12 x)^(1/2), {x, 0, 19}], x] (* _Michael De Vlieger_, Aug 26 2015 *)
%t Join[{1}, RecurrenceTable[{a[1] == 6, a[n] == a[n-1] (18/n - 12)}, a, {n, 20}]] (* _Vincenzo Librandi_, Aug 27 2015 *)
%o (PARI) x = xx+O(xx^30); Vec(sqrt(1 + 12*x)) \\ _Michel Marcus_, Aug 26 2015
%o (Magma) [1] cat [(2/3)*(-3)^(n+1)*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, May 21 2022
%o (SageMath) [(2/3)*(-3)^(n+1)*catalan_number(n-1) for n in (0..30)] # _G. C. Greubel_, May 21 2022
%Y Cf. A000108, A025226.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 22 2005
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