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a(n) = denominator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).
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%I #11 Aug 17 2022 22:30:29

%S 1,1,1,3,12,60,180,630,10080,18144,453600,2494800,59875200,778377600,

%T 1089728640,40864824000,1307674368000,22230464256000,15390321408000,

%U 380140938777600,76028187755520000,1596591942865920000

%N a(n) = denominator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

%C Sum_{k >= 1} b(k) = e^(3/2) where e = 2.718... . More generally if b(1) = b(2) = ... = b(m) = 1 and b(n+m+1) = 1/(n+m)*( b(n+m) + b(n+m-1) + ... + b(n) ) then Sum_{k >= 1} b(k) = e^H(m) where H(m) = Sum_{j=1..m} 1/j is the m-th harmonic number (Benoit Cloitre and Boris Gourevitch).

%H G. C. Greubel, <a href="/A069944/b069944.txt">Table of n, a(n) for n = 1..350</a>

%F A069943(n)/a(n) = A000085(n-1)/A000142(n-1) in lowest terms. - _Christian G. Bower_, Jan 14 2006

%F a(n) = denominator( A013989(n-1)/n! ). - _G. C. Greubel_, Aug 17 2022

%t Table[Denominator[n*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I/Sqrt[2]]/n!], {n, 30}] (* _G. C. Greubel_, Aug 17 2022 *)

%o (Magma)

%o A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;

%o A069944:= func< n | Denominator(A013989(n-1)/Factorial(n-1)) >;

%o [A069944(n): n in [1..30]]; // _G. C. Greubel_, Aug 17 2022

%o (SageMath)

%o @CachedFunction

%o def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n

%o [denominator(A013989(n-1)/factorial(n)) for n in (1..30)] # _G. C. Greubel_, Aug 17 2022

%Y Cf. A000085, A000142, A013989, A069943.

%K easy,frac,nonn

%O 1,4

%A _Benoit Cloitre_, Apr 27 2002