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A304312 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.

%I A304312 #24 Feb 27 2021 05:36:34
%S A304312 1,9,148,3493,106431,3950832,172325014,8617033285,485267003023,
%T A304312 30363691715629,2088698040637242,156612539215405732,
%U A304312 12709745319947141220,1109746209390479579732,103724343230007402591558,10332348604630683943445797,1092720669631704348689818959,122274820828415241343176467043,14433472319311799728710020346232
%N A304312 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.
%C A304312 Is this sequence essentially the same as A006691?
%C A304312 Conjecture: a(n) is the number of connected n-state finite automata with 2 inputs (A006691). [I believe the name of A006691 should be changed to read "(n+1)-state". See my comments in A006691. - _Petros Hadjicostas_, Feb 26 2021]
%C A304312 Equals row 2 of table A304321.
%H A304312 Paul D. Hanna, <a href="/A304312/b304312.txt">Table of n, a(n) for n = 0..400</a>
%F A304312 Logarithmic derivative of the o.g.f. of A304322.
%F A304312 For n>=1, a(n) = B_{n+1}((n+1)^2-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - _Max Alekseyev_, Jun 18 2018
%F A304312 a(n) ~ sqrt(1-c) * 2^(2*n + 3/2) * n^(n + 3/2) / (sqrt(Pi) * c^(n+1) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - _Vaclav Kotesovec_, Aug 31 2020
%e A304312 O.g.f.: L(x) = 1 + 9*x + 148*x^2 + 3493*x^3 + 106431*x^4 + 3950832*x^5 + 172325014*x^6 + 8617033285*x^7 + 485267003023*x^8 + 30363691715629*x^9 + ...
%e A304312 such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304322 :
%e A304312 F(x) = 1 + x + 5*x^2 + 54*x^3 + 935*x^4 + 22417*x^5 + 685592*x^6 + 25431764*x^7 + 1106630687*x^8 + 55174867339*x^9 + 3097872254493*x^10 + ... + A304322(n)*x^n + ...
%e A304312 which satisfies [x^n] exp( n^2 * x ) / F(x) = 0 for n>0.
%t A304312 m = 25;
%t A304312 F = 1 + Sum[c[k] x^k, {k, m}];
%t A304312 s[n_] := Solve[SeriesCoefficient[Exp[n^2 * x]/F, {x, 0, n}] == 0][[1]];
%t A304312 Do[F = F /. s[n], {n, m}];
%t A304312 CoefficientList[D[F, x]/F + O[x]^m, x] (* _Jean-François Alcover_, May 20 2018 *)
%o A304312 (PARI) {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^2 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
%o A304312 for(n=0,25, print1( a(n),", "))
%Y A304312 Cf. A304322, A006691, A304321, A304313, A304314, A304315.
%K A304312 nonn
%O A304312 0,2
%A A304312 _Paul D. Hanna_, May 11 2018