A069943 a(n) = numerator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).

1, 1, 1, 2, 5, 13, 19, 29, 191, 131, 1187, 2231, 17519, 71063, 29881, 323423, 2887921, 13237457, 2397389, 15030317, 742458253, 3748521653, 9670072483, 25451905333, 10932619111, 78684575461, 4163946939067, 11799518538967, 136025604432743

Offset: 1

Author

Benoit Cloitre, Apr 27 2002

Keywords

• easy
• frac
• nonn

Comments

Sum_{k>=1} b(k) = exp(3/2). 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) = exp(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..840

Crossrefs

Cf. A000085, A000142, A013989, A069944.

Programs

Magma

A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;
A069944:= func< n | Numerator(A013989(n-1)/Factorial(n)) >;
[A069944(n): n in [1..40]]; // G. C. Greubel, Aug 17 2022

Mathematica

Table[Numerator[n*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I/Sqrt[2]]/n!], {n, 40}] (* G. C. Greubel, Aug 17 2022 *)
nxt[{n_,a_,b_}]:={n+1,b,(a+b)/n}; NestList[nxt,{2,1,1},30][[;;,2]]//Numerator (* Harvey P. Dale, Feb 02 2025 *)

SageMath

@CachedFunction
def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n
[numerator(A013989(n-1)/factorial(n)) for n in (1..40)] # G. C. Greubel, Aug 17 2022

Formula

a(n)/A069944(n) = A000085(n-1)/A000142(n-1) in lowest terms. [Christian G. Bower, Jan 14 2006]

Numerators in the power series of exp(x+x^2/2) (e.g.f. for involutions, cf. A000085). exp(x+x^2/2) = 1 + x + x^2 + 2/3*x^3 + 5/12*x^4 + 13/60*x^5 + 19/180*x^6 + 29/630*x^7 + 191/10080*x^8 + ... [Joerg Arndt, May 10 2008]

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