A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.

1, 1, 25, 2317, 466241, 162016980, 85975473871, 64545532370208, 65062315637060121, 84756897268784533255, 138581022247955235150982, 277878562828788369685779910, 670574499099019193091230751539, 1917288315895234006935990419270242, 6409780596355519454337664637246378856, 24774712941456386970945752104780461007848, 109632095120643795798521114315908854415860345

Offset

0,3

Comments

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

Equals row 3 of table A304320.

O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107675.

Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304312.

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 3 inputs (A006692).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n) ~ sqrt(1-c) * 3^(3*n) * n^(2*n - 1/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n) * exp(2*n)), where c = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Aug 31 2020

Example

O.g.f.: A(x) = 1 + x + 25*x^2 + 2317*x^3 + 466241*x^4 + 162016980*x^5 + 85975473871*x^6 + 64545532370208*x^7 + 65062315637060121*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x) begins:
n=0: [1, -1, -48, -13608, -11065344, -19317285000, -61649646030720, ...];
n=1: [1, 0, -49, -13754, -11120067, -19372748284, -61765715993765, ...];
n=2: [1, 7, 0, -14440, -11517184, -19768841352, -62587640670464, ...];
n=3: [1, 26, 627, 0, -12292251, -20908064898, -64905483973113, ...];
n=4: [1, 63, 3920, 227032, 0, -22551552136, -69768485886848, ...];
n=5: [1, 124, 15327, 1874642, 213958781, 0, -75806801733845, ...];
n=6: [1, 215, 46176, 9893016, 2100211968, 416846973816, 0, ...];
n=7: [1, 342, 116915, 39937660, 13616254341, 4604681316698, 1458047845980391, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^3*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304313:
A'(x)/A(x) = 1 + 49*x + 6877*x^2 + 1854545*x^3 + 807478656*x^4 + 514798204147*x^5 + 451182323794896*x^6 + 519961864703259753*x^7 + ... + A304313(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107675:
B(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + ... + A107675(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^3 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

Cf. A304320, A304313, A304321, A304322, A304324, A304325.

Cf. A006691, A107675.

Sequence in context: A266130 A363192 A222276 * A135057 A324417 A167036

Adjacent sequences: A304320 A304321 A304322 * A304324 A304325 A304326

Keyword

nonn

Author

Paul D. Hanna, May 11 2018

Status

approved