A069944 a(n) = denominator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

1, 1, 1, 3, 12, 60, 180, 630, 10080, 18144, 453600, 2494800, 59875200, 778377600, 1089728640, 40864824000, 1307674368000, 22230464256000, 15390321408000, 380140938777600, 76028187755520000, 1596591942865920000

Offset: 1

Author

Benoit Cloitre, Apr 27 2002

Keywords

• easy
• frac
• nonn

Comments

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).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000085, A000142, A013989, A069943.

Programs

Magma

A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;
A069944:= func< n | Denominator(A013989(n-1)/Factorial(n-1)) >;
[A069944(n): n in [1..30]]; // G. C. Greubel, Aug 17 2022

Mathematica

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

SageMath

@CachedFunction
def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n
[denominator(A013989(n-1)/factorial(n)) for n in (1..30)] # G. C. Greubel, Aug 17 2022

Formula

A069943(n)/a(n) = A000085(n-1)/A000142(n-1) in lowest terms. - Christian G. Bower, Jan 14 2006

a(n) = denominator( A013989(n-1)/n! ). - G. C. Greubel, Aug 17 2022