A304324 - OEIS

%I #14 Aug 31 2020 08:53:09

%S 1,1,113,76446,153143499,673638499100,5510097691767062,

%T 75312181798660695788,1595682359653020033714019,

%U 49564410138113345565513815041,2161639124039437373346491749452440,127889301139607880711208251726358504898,9979766671875039854419652569806336108694074

%N O.g.f. A(x) satisfies: [x^n] exp( n^4 * 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 4 of table A304320.

%C O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304394.

%C Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304314.

%C Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) is the number of connected n-state finite automata with 4 inputs.

%H Paul D. Hanna, <a href="/A304324/b304324.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ sqrt(1-c) * 4^(4*n) * n^(3*n - 1/2) / (sqrt(2*Pi) * c^n * (4-c)^(3*n) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - _Vaclav Kotesovec_, Aug 31 2020

%e O.g.f.: A(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp(n^4*x) / A(x) begins:

%e n=0: [1, -1, -224, -457326, -3671476224, -80797824300000, ...];

%e n=1: [1, 0, -225, -458000, -3673306875, -80816186256624, ...];

%e n=2: [1, 15, 0, -464750, -3701040000, -81092721606624, ...];

%e n=3: [1, 80, 6175, 0, -3787546875, -82312696206624, ...];

%e n=4: [1, 255, 64800, 15951250, 0, -84756571206624, ...];

%e n=5: [1, 624, 389151, 242091424, 146271536901, 0, ...];

%e n=6: [1, 1295, 1676800, 2170415250, 2804103120000, 3524906587193376, 0, ...]; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^4*x ) / A(x) = 0 for n>=0.

%e LOGARITHMIC DERIVATIVE.

%e The logarithmic derivative of A(x) yields the o.g.f. of A304314:

%e A'(x)/A(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ... + A304314(n)*x^n + ...

%e INVERT TRANSFORM.

%e 1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304394:

%e B(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ... + A304394(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)^4 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

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

%Y Cf. A304320, A304314, A304321, A304322, A304323, A304325.

%Y Cf. A304394.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 11 2018