|
%I #6 Mar 21 2018 07:51:23
%S 1,2,10,78,802,10058,147442,2461054,45960098,948268402,21410711450,
%T 525049525294,13897732641954,394987936658714,11999174713271266,
%U 388077151776127486,13315213471551257154,483131189591348032482,18485324379022683692714,743888762544523242047886,31411035576323146658185122,1388712621964856674998780010,64156199255423145619052883154,3091505922381615544789816776830
%N G.f. A(x) satisfies: A(x) = x * (1 + x*A'(x)) / (1 - x*A'(x)).
%C O.g.f. equals the logarithm of the e.g.f. of A301387.
%C The e.g.f. G(x) of A301387 satisfies: [x^n] G(x)^n = (2*n - 1) * [x^(n-1)] G(x)^n for n>=1.
%H Paul D. Hanna, <a href="/A301388/b301388.txt">Table of n, a(n) for n = 1..300</a>
%F O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = (2*n - 1) * [x^(n-1)] exp( n * A(x) ) for n>=1.
%F a(n) ~ c * 2^n * n!, where c = 0.321697697353832218399635... - _Vaclav Kotesovec_, Mar 21 2018
%e G.f.: A(x) = x + 2*x^2 + 10*x^3 + 78*x^4 + 802*x^5 + 10058*x^6 + 147442*x^7 + 2461054*x^8 + 45960098*x^9 + 948268402*x^10 + ...
%e where
%e A(x) = x*(1 + x*A'(x)) / (1 - x*A'(x)).
%e RELATED SERIES.
%e A'(x) = 1 + 4*x + 30*x^2 + 312*x^3 + 4010*x^4 + 60348*x^5 + 1032094*x^6 + 19688432*x^7 + 413640882*x^8 + ...
%e exp(A(x)) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2185*x^4/4! + 108881*x^5/5! + 8012941*x^6/6! + 809101945*x^7/7! + 106751544593*x^8/8! + ... + A301387*x^n/n! + ...
%o (PARI) {a(n) = my(A=x); for(i=0, n, A = x*(1 + x*A')/(1 - x*A' +x*O(x^n)) ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A301387.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 20 2018
| 