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

%I A069943 #18 Feb 02 2025 13:11:56
%S A069943 1,1,1,2,5,13,19,29,191,131,1187,2231,17519,71063,29881,323423,
%T A069943 2887921,13237457,2397389,15030317,742458253,3748521653,9670072483,
%U A069943 25451905333,10932619111,78684575461,4163946939067,11799518538967,136025604432743
%N A069943 a(n) = numerator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).
%C A069943 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]
%H A069943 G. C. Greubel, <a href="/A069943/b069943.txt">Table of n, a(n) for n = 1..840</a>
%F A069943 a(n)/A069944(n) = A000085(n-1)/A000142(n-1) in lowest terms. [_Christian G. Bower_, Jan 14 2006]
%F A069943 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]
%F A069943 a(n) = numerator( A013989(n-1)/n! ). - _G. C. Greubel_, Aug 17 2022
%t A069943 Table[Numerator[n*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I/Sqrt[2]]/n!], {n, 40}] (* _G. C. Greubel_, Aug 17 2022 *)
%t A069943 nxt[{n_,a_,b_}]:={n+1,b,(a+b)/n}; NestList[nxt,{2,1,1},30][[;;,2]]//Numerator (* _Harvey P. Dale_, Feb 02 2025 *)
%o A069943 (Magma)
%o A069943 A013989:= func< n | (&+[Factorial(n)/(2^k*Factorial(n-2*k)*Factorial(k)): k in [0..Floor(n/2)]]) >;
%o A069943 A069944:= func< n | Numerator(A013989(n-1)/Factorial(n)) >;
%o A069943 [A069944(n): n in [1..40]]; // _G. C. Greubel_, Aug 17 2022
%o A069943 (SageMath)
%o A069943 @CachedFunction
%o A069943 def A013989(n): return n+1 if (n<2) else (n+1)*(A013989(n-1) + n*A013989(n-2))/n
%o A069943 [numerator(A013989(n-1)/factorial(n)) for n in (1..40)] # _G. C. Greubel_, Aug 17 2022
%Y A069943 Cf. A000085, A000142, A013989, A069944.
%K A069943 easy,frac,nonn
%O A069943 1,4
%A A069943 _Benoit Cloitre_, Apr 27 2002