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A176730
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Denominators of coefficients of a series, called f, related to Airy functions.
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5
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1, 6, 180, 12960, 1710720, 359251200, 109930867200, 46170964224000, 25486372251648000, 17891433320656896000, 15565546988971499520000, 16437217620353903493120000, 20710894201645918401331200000, 30693545206839251070772838400000, 52854284846177190343870827724800000
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OFFSET
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0,2
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COMMENTS
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The numerators are always 1.
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.
If y = Sum_{n >= 0} x^(3*n)/a(n), then y'' = x*y. - Michael Somos, Jul 12 2019
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
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
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, 10.4.2 - 5. [alternative scanned copy].
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FORMULA
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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.
a(n) = 3*n*(3*n - 1)*a(n-1) with a(0) = 1.
a(n) = (3*n + 1)!/(n!*3^n)*Sum_{k = 0..n} (-1)^k*binomial(n,k)/(3*k + 1).
a(n) = (3*n + 1)!/(n!*3^n)*hypergeom([-n, 1/3], [4/3], 1).
a(n) = (2*Pi*sqrt(3))/9 * 1/(3^n) * Gamma(3*n+2)/(Gamma(2/3)*Gamma(n+4/3)).
(End)
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EXAMPLE
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Rational f-coefficients: 1, 1/6, 1/180, 1/12960, 1/1710720, 1/359251200, 1/109930867200, 1/46170964224000, ....
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MAPLE
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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
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MATHEMATICA
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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 *)
a[ n_] := If[ n < 0, 0, (3*n)! / Product[ k, {k, 1, 3*n - 2, 3}]]; (* Michael Somos, Oct 14 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, (3*n)! / prod( k=0, n-1, 3*k + 1))}; /* Michael Somos, Oct 14 2011 */
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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