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 A304314 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0. 6
 1, 225, 229000, 612243125, 3367384031526, 33056423981177346, 527146092112494861420, 12764850938355048224394925, 446065249480005516657138106375, 21615893741029073481369412949207860, 1406758471936562034421316174257309550136, 119755662436589797897149020637183902177930534 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: a(n) is the number of connected n-state finite automata with 4 inputs. Equals row 4 of table A304321. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA Logarithmic derivative of the o.g.f. of A304324. For n>=1, a(n) = B_{n+1}((n+1)^4-0!*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 a(n) ~ sqrt(1-c) * 4^(4*(n+1)) * n^(3*n + 7/2) / (sqrt(2*Pi) * c^(n+1) * (4-c)^(3*(n+1)) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - Vaclav Kotesovec, Aug 31 2020 EXAMPLE O.g.f.: L(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ... such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 : F(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ... + A304324(n)*x^n + ... which satisfies [x^n] exp( n^4 * x ) / F(x) = 0 for n>0. MATHEMATICA m = 25; F = 1 + Sum[c[k] x^k, {k, m}]; s[n_] := Solve[SeriesCoefficient[Exp[n^4*x]/F, {x, 0, n}] == 0][[1]]; Do[F = F /. s[n], {n, m}]; CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *) PROG (PARI) {a(n) = my(A=[1], L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]} for(n=0, 25, print1( a(n), ", ")) CROSSREFS Cf. A304324, A304321, A304312, A304313, A304315. Sequence in context: A265420 A171109 A239478 * A109688 A195277 A319944 Adjacent sequences:  A304311 A304312 A304313 * A304315 A304316 A304317 KEYWORD nonn AUTHOR Paul D. Hanna, May 11 2018 STATUS approved

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Last modified January 16 14:39 EST 2022. Contains 350376 sequences. (Running on oeis4.)