A176730 - OEIS

%I #65 Nov 17 2024 03:18:34

%S 1,6,180,12960,1710720,359251200,109930867200,46170964224000,

%T 25486372251648000,17891433320656896000,15565546988971499520000,

%U 16437217620353903493120000,20710894201645918401331200000,30693545206839251070772838400000,52854284846177190343870827724800000

%N Denominators of coefficients of a series, called f, related to Airy functions.

%C The numerators are always 1.

%C Let f(z) = Sum_{n>=0} (1/a(n))*z^(3*n) and g(z) = Sum_{n>=0}(1/b(n))*z^(3*n+1) with b(n) = A176731(n) build the two independent Airy functions Ai(z) = c[1]*f(z) - c[2]*g(z) and Bi(z) = sqrt(3)*(c[1]*f(z) + c[2]*g(z)) with c[1] = 1/(3^(2/3)*Gamma(2/3)), approximately 0.35502805388781723926 and c[2] = 1/(3^(1/3)*Gamma(1/3)), approximately 0.25881940379280679840.

%C If y = Sum_{n >= 0} x^(3*n)/a(n), then y'' = x*y. - _Michael Somos_, Jul 12 2019

%C Define W(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(a(n)*(3*n+1)). Then W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. The function 1/W(z) is the e.g.f. of A117226, which is the number of permutations of [n] avoiding the consecutive pattern 1243. In other words, Sum_{n >= 0} A117226(n)*z^n/n! = 1/W(z). See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003). - _Petros Hadjicostas_, Nov 01 2019

%C If y = Sum_{n >= 0} a(n)*x^(3*n+1)/(3*n+1)!, then y' = 1 + x^2*y. - _Michael Somos_, May 22 2022

%H G. C. Greubel, <a href="/A176730/b176730.txt">Table of n, a(n) for n = 0..200</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, 10.4.2 - 5. [alternative scanned copy].

%H Sergi Elizalde and Marc Noy, <a href="https://doi.org/10.1016/S0196-8858(02)00527-4 ">Consecutive patterns in permutations</a>, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.

%H Wolfdieter Lang, <a href="/A176730/a176730.txt">The first 20 terms of the f(z) and g(z) functions</a>.

%H NIST's Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/9.4">Airy and Related Functions (Maclaurin Series)</a> by Frank W. J. Olver.

%F a(n) = denominator((3^n)*risefac(1/3,n)/(3*n)!) with the rising factorials risefac(k,n) = Product_{j=0..n-1} (k+j) and risefac(k,0)=1.

%F From _Peter Bala_, Dec 13 2021: (Start)

%F a(n) = 3*n*(3*n - 1)*a(n-1) with a(0) = 1.

%F a(n) = (3*n + 1)!/(n!*3^n)*Sum_{k = 0..n} (-1)^k*binomial(n,k)/(3*k + 1).

%F a(n) = (3*n + 1)!/(n!*3^n)*hypergeom([-n, 1/3], [4/3], 1).

%F a(n) = (2*Pi*sqrt(3))/9 * 1/(3^n) * Gamma(3*n+2)/(Gamma(2/3)*Gamma(n+4/3)).

%F (End)

%F a(n) = (9^n*n!*(n-1/3)!)/(-1/3)!. - _Peter Luschny_, Dec 20 2021

%F a(n) = A014402(2*n). - _Michael Somos_, May 22 2022

%e Rational f-coefficients: 1, 1/6, 1/180, 1/12960, 1/1710720, 1/359251200, 1/109930867200, 1/46170964224000, ....

%p a := proc (n) option remember; if n = 0 then 1 else 3*n*(3*n-1)*a(n-1) end if; end proc: seq(a(n), n = 0..20); # _Peter Bala_, Dec 13 2021

%t a[ n_] := If[ n < 0, 0, 1 / (3^(2/3) Gamma[2/3] SeriesCoefficient[ AiryAi[x], {x, 0, 3*n}])]; (* _Michael Somos_, Oct 14 2011 *)

%t a[ n_] := If[ n < 0, 0, (3*n)! / Product[ k, {k, 1, 3*n - 2, 3}]]; (* _Michael Somos_, Oct 14 2011 *)

%o (PARI) {a(n) = if( n<0, 0, (3*n)! / prod( k=0, n-1, 3*k + 1))}; /* _Michael Somos_, Oct 14 2011 */

%Y Cf. A014402, A117226, A176731.

%Y Column k=3 of A329070.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 14 2010