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%I #28 Sep 20 2023 01:45:39
%S 1,1,6,12,180,504,12960,45360,1710720,7076160,359251200,1698278400,
%T 109930867200,580811212800,46170964224000,268334780313600,
%U 25486372251648000,161000868188160000,17891433320656896000
%N Numbers found in denominators of expansion of Airy function Ai(x).
%C Although the description is technically correct, this sequence is unsatisfactory because there are gaps in the series.
%C A014402 arises via Vandermonde determinants as in A203433; see the Mathematica section. - _Clark Kimberling_, Jan 02 2012
%H G. C. Greubel, <a href="/A014402/b014402.txt">Table of n, a(n) for n = 0..420</a>
%H NIST's Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/Contents/AI/AI.4.html#eq:AI.MC.AI">Airy and Related Functions (Maclaurin Series)</a> by Frank W. J. Olver.
%F a(2*n) = A176730(n). a(2*n + 1) = A176731(n). - _Michael Somos_, Oct 14 2011
%e Mathematica gives the series as 1/(3^(2/3)*Gamma(2/3)) - x/(3^(1/3)*Gamma(1/3)) + x^3/(6*3^(2/3)*Gamma(2/3)) - x^4/(12*3^(1/3)*Gamma(1/3) + x^6/(180*3^(2/3)*Gamma(2/3) - x^7/(504*3^(1/3)*Gamma(1/3) + x^9/(12960*3^(2/3)*Gamma(2/3) - ...
%t Series[ AiryAi[ x ], {x, 0, 30} ]
%t a[ n_] := If[ n<0, 0, (n + Quotient[ n, 2])! / Product[ 3 k + 1 + Mod[n, 2], {k, 0, Quotient[ n, 2] - 1}]]; (* _Michael Somos_, Oct 14 2011 *)
%t (* Next, A014402 generated in via Vandermonde determinants based on A007494 *)
%t f[j_]:= j + Floor[(j+1)/2]; z = 20;
%t v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t d[n_]:= Product[(i-1)!, {i,n}]
%t Table[v[n], {n,z}] (* A203433 *)
%t Table[v[n+1]/v[n], {n,z}] (* this sequence *)
%t Table[v[n]/d[n], {n,z}] (* A203434 *)
%t (* _Clark Kimberling_, Jan 02 2012 *)
%o (PARI) {a(n) = if( n<0, 0, (n\2 + n)! / prod( k=0, n\2 -1, n%2 + 3*k + 1))}; /* _Michael Somos_, Oct 14 2011 */
%o (Magma)
%o A014402:= func< n | n eq 0 select 1 else (&*[n-j+Floor(n/2)-Floor(j/2): j in [0..n-1]]) >;
%o [A014402(n): n in [0..25]]; // _G. C. Greubel_, Sep 20 2023
%o (SageMath)
%o def A014402(n): return product(n-j+(n//2)-(j//2) for j in range(n))
%o [A014402(n) for n in range(31)] # _G. C. Greubel_, Sep 20 2023
%Y Cf. A014403, A060507, A176730, A176731, A203433, A203434.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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