A304325 - OEIS

%I #16 Sep 21 2022 23:27:27

%S 1,1,481,2246281,43087884081,2331601789103231,287133439746933073357,

%T 69929721774643572422651223,30496192503451926066104677123329,

%U 22113985380962062942048847693898939310,25177466100486219354624677349405490885006591,42994825404638061265611776726882581676486680632128

%N O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) / A(x) = 0 for n>0.

%C It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.

%C Equals row 5 of table A304320.

%C O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304395.

%C Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304315.

%C Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) enumerates the connected n-state finite automata with 5 inputs.

%H Paul D. Hanna, <a href="/A304325/b304325.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ sqrt(1-c) * 5^(5*n) * n^(4*n - 1/2) / (sqrt(2*Pi) * c^n * (5-c)^(4*n) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - _Vaclav Kotesovec_, Aug 31 2020

%e O.g.f.: A(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp(n^5*x) / A(x) begins:

%e n=0: [1, -1, -960, -13471920, -1033995878400, -279781615181250000, ...];

%e n=1: [1, 0, -961, -13474802, -1034049771843, -279786785295370804, ...];

%e n=2: [1, 31, 0, -13534384, -1035725264896, -279947192760516048, ...];

%e n=3: [1, 242, 57603, 0, -1044001318107, -281045183102366562, ...];

%e n=4: [1, 1023, 1045568, 1054175056, 0, -284106842971323856, ...];

%e n=5: [1, 3124, 9758415, 30465809330, 93986716449725, 0, ...];

%e n=6: [1, 7775, 60449664, 469967719248, 3652476388472832, 28079364132086235696, 0, ...]; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^5*x ) / A(x) = 0 for n >= 0.

%e LOGARITHMIC DERIVATIVE.

%e The logarithmic derivative of A(x) yields the o.g.f. of A304315:

%e A'(x)/A(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ... + A304315(n)*x^n +...

%e INVERT TRANSFORM.

%e 1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304395:

%e B(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + ... + A304395(n)*x^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^5 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

%o for(n=0,25, print1( a(n),", "))

%Y Cf. A304320, A304315, A304321, A304322, A304323, A304324.

%Y Cf. A304395.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 11 2018