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%I #26 Aug 31 2020 10:58:35
%S 1,1,1,1,9,1,1,49,148,1,1,225,6877,3493,1,1,961,229000,1854545,106431,
%T 1,1,3969,6737401,612243125,807478656,3950832,1,1,16129,188580028,
%U 172342090401,3367384031526,514798204147,172325014,1,1,65025,5170118437,45770504571813,11657788116175751,33056423981177346,451182323794896,8617033285,1
%N Table of coefficients in row functions F'(n,x)/F(n,x) such that [x^k] exp( k^n * x ) / F(n,x) = 0 for k>=1 and n>=1.
%C Conjecture: T(n,k) in row n and column k gives the number of connected k-state finite automata with n inputs, for k>=0, for n>=1. For example, row 2 agrees with A006691, the number of connected n-state finite automata with 2 inputs; also, row 3 agrees with A006692, the number of connected n-state finite automata with 3 inputs.
%H Paul D. Hanna, <a href="/A304321/b304321.txt">Table of n, a(n) for n = 1..1326 as a flattened table read by antidiagonals 1..51.</a>
%F Row n of this table equals the logarithmic derivative of row n of table A304320.
%F For fixed row r > 1 is a(n) ~ sqrt(1-c) * r^(r*(n+1)) * n^((r-1)*n + r - 1/2) / (sqrt(2*Pi) * c^(n+1) * (r-c)^((r-1)*(n+1)) * exp((r-1)*n)), where c = -LambertW(-r*exp(-r)). - _Vaclav Kotesovec_, Aug 31 2020
%e This table begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e 1, 9, 148, 3493, 106431, 3950832, 172325014, 8617033285, 485267003023, ...;
%e 1, 49, 6877, 1854545, 807478656, 514798204147, 451182323794896, ...;
%e 1, 225, 229000, 612243125, 3367384031526, 33056423981177346, ...;
%e 1, 961, 6737401, 172342090401, 11657788116175751, 1722786509653595220757, ...;
%e 1, 3969, 188580028, 45770504571813, 37854124915368647781, ...;
%e 1, 16129, 5170118437, 11889402239702065, 120067639589726126102806, ...;
%e 1, 65025, 140510362000, 3061712634885743125, 377436820462509018320487276, ...;
%e 1, 261121, 3804508566001, 785701359968473902401, 1182303741240112494973150131501, ...; ...
%e Let F'(n,x)/F(n,x) denote the o.g.f. of row n of this table, then the coefficient of x^k in exp(k^n*x)/F(n,x) = 0 for k>=1 and n>=1.
%t m = 10(*rows*);
%t row[nn_] := Module[{F, s}, F = 1 + Sum[c[k] x^k, {k, m}]; s[n_] := Solve[ SeriesCoefficient[Exp[n^nn*x]/F, {x, 0, n}] == 0][[1]]; Do[F = F /. s[n], {n, m}]; CoefficientList[D[F, x]/F + O[x]^m, x]];
%t T = Array[row, m];
%t Table[T[[n-k+1, k]], {n, 1, m}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 27 2019 *)
%o (PARI) {T(n,k) = my(A=[1],m); for(i=0, k, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^n +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[k+1]}
%o /* Print table: */
%o for(n=1,8, for(k=0,8, print1( T(n,k),", "));print(""))
%o /* Print as a flattened table: */
%o for(n=0,10, for(k=0,n, print1( T(n-k+1,k),", "));)
%Y Cf. A304320, A304312 (row 2), A304313 (row 3), A304314 (row 4), A304315 (row 5).
%K nonn,tabl
%O 1,5
%A _Paul D. Hanna_, May 11 2018
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