A108735 Expansion of sqrt(1 + 12*x).

1, 6, -18, 108, -810, 6804, -61236, 577368, -5629338, 56293380, -574192476, 5950722024, -62482581252, 663276631752, -7106535340200, 76750581674160, -834662575706490, 9132190534200420, -100454095876204620, 1110282112315945800

Offset

0,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..838

Formula

From Wolfdieter Lang, Aug 26 2015: (Start)

G.f.: sqrt(1 + 12*x) = 1 + 6*x*c(-3*x), with the g.f. c of the Catalan numbers A000108.

a(n) = -2*(-3)^n*C(n-1), n >= 1, and a(0) = 1, with C(n) = A000108(n). (End)

From Robert Israel, Aug 27 2015: (Start)

D-finite with recurrence: a(n) = (18/n - 12)*a(n-1).

a(n) ~ (-1)^(n+1)*12^n/(2*sqrt(Pi)*n^(3/2)). (End)

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

a(n) = 2*(-1)^(n+1)*A025226(n). - R. J. Mathar, Jan 23 2020

From Amiram Eldar, May 28 2022: (Start)

Sum_{n>=0} 1/a(n) = (192 - 36*arcsinh(1/(2*sqrt(3)))/sqrt(13))/169.

Sum_{n>=0} (-1)^n/a(n) = (96 - 36*arcsin(1/(2*sqrt(3)))/sqrt(11))/121.

Example

G.f. = 1 + 6*x - 18*x^2 + 108*x^3 - 810*x^4 + 6804*x^5 - 61236*x^6 + ...

Maple

f:= proc(n) option remember; (18/n - 12)*procname(n-1) end proc: f(0):= 1:
map(f, [$0..100]); # Robert Israel, Aug 27 2015

Mathematica

CoefficientList[Series[(1 + 12 x)^(1/2), {x, 0, 19}], x] (* Michael De Vlieger, Aug 26 2015 *)
Join[{1}, RecurrenceTable[{a[1] == 6, a[n] == a[n-1] (18/n - 12)}, a, {n, 20}]] (* Vincenzo Librandi, Aug 27 2015 *)

Prog

(PARI) x = xx+O(xx^30); Vec(sqrt(1 + 12*x)) \\ Michel Marcus, Aug 26 2015
(Magma) [1] cat [(2/3)*(-3)^(n+1)*Catalan(n-1): n in [1..30]]; // G. C. Greubel, May 21 2022
(SageMath) [(2/3)*(-3)^(n+1)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, May 21 2022

Crossrefs

Cf. A000108, A025226.

Sequence in context: A222857 A367664 A361729 * A143556 A007126 A009576

Adjacent sequences: A108732 A108733 A108734 * A108736 A108737 A108738

Keyword

sign,easy

Author

N. J. A. Sloane, Jun 22 2005

Status

approved