A304321 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.

1, 1, 1, 1, 9, 1, 1, 49, 148, 1, 1, 225, 6877, 3493, 1, 1, 961, 229000, 1854545, 106431, 1, 1, 3969, 6737401, 612243125, 807478656, 3950832, 1, 1, 16129, 188580028, 172342090401, 3367384031526, 514798204147, 172325014, 1, 1, 65025, 5170118437, 45770504571813, 11657788116175751, 33056423981177346, 451182323794896, 8617033285, 1

Offset

1,5

Comments

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.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1326 as a flattened table read by antidiagonals 1..51.

Formula

Row n of this table equals the logarithmic derivative of row n of table A304320.

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

Example

This table begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 9, 148, 3493, 106431, 3950832, 172325014, 8617033285, 485267003023, ...;
1, 49, 6877, 1854545, 807478656, 514798204147, 451182323794896, ...;
1, 225, 229000, 612243125, 3367384031526, 33056423981177346, ...;
1, 961, 6737401, 172342090401, 11657788116175751, 1722786509653595220757, ...;
1, 3969, 188580028, 45770504571813, 37854124915368647781, ...;
1, 16129, 5170118437, 11889402239702065, 120067639589726126102806, ...;
1, 65025, 140510362000, 3061712634885743125, 377436820462509018320487276, ...;
1, 261121, 3804508566001, 785701359968473902401, 1182303741240112494973150131501, ...; ...
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.

Mathematica

m = 10(*rows*);
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 = Array[row, m];
Table[T[[n-k+1, k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 27 2019 *)

Prog

(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]}
/* Print table: */
for(n=1, 8, for(k=0, 8, print1( T(n, k), ", ")); print(""))
/* Print as a flattened table: */
for(n=0, 10, for(k=0, n, print1( T(n-k+1, k), ", ")); )

Crossrefs

Cf. A304320, A304312 (row 2), A304313 (row 3), A304314 (row 4), A304315 (row 5).

Sequence in context: A181144 A142468 A359313 * A156278 A166961 A202988

Adjacent sequences: A304318 A304319 A304320 * A304322 A304323 A304324

Keyword

nonn,tabl

Author

Paul D. Hanna, May 11 2018

Status

approved