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

%I A304323 #19 Aug 31 2020 08:06:33
%S A304323 1,1,25,2317,466241,162016980,85975473871,64545532370208,
%T A304323 65062315637060121,84756897268784533255,138581022247955235150982,
%U A304323 277878562828788369685779910,670574499099019193091230751539,1917288315895234006935990419270242,6409780596355519454337664637246378856,24774712941456386970945752104780461007848,109632095120643795798521114315908854415860345
%N A304323 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x ) / A(x) = 0 for n>0.
%C A304323 It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
%C A304323 Equals row 3 of table A304320.
%C A304323 O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107675.
%C A304323 Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304312.
%C A304323 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).
%H A304323 Paul D. Hanna, <a href="/A304323/b304323.txt">Table of n, a(n) for n = 0..300</a>
%F A304323 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
%e A304323 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 + ...
%e A304323 ILLUSTRATION OF DEFINITION.
%e A304323 The table of coefficients of x^k/k! in exp(n^3*x) / A(x) begins:
%e A304323 n=0: [1, -1, -48, -13608, -11065344, -19317285000, -61649646030720, ...];
%e A304323 n=1: [1, 0, -49, -13754, -11120067, -19372748284, -61765715993765, ...];
%e A304323 n=2: [1, 7, 0, -14440, -11517184, -19768841352, -62587640670464, ...];
%e A304323 n=3: [1, 26, 627, 0, -12292251, -20908064898, -64905483973113, ...];
%e A304323 n=4: [1, 63, 3920, 227032, 0, -22551552136, -69768485886848, ...];
%e A304323 n=5: [1, 124, 15327, 1874642, 213958781, 0, -75806801733845, ...];
%e A304323 n=6: [1, 215, 46176, 9893016, 2100211968, 416846973816, 0, ...];
%e A304323 n=7: [1, 342, 116915, 39937660, 13616254341, 4604681316698, 1458047845980391, 0, ...]; ...
%e A304323 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.
%e A304323 LOGARITHMIC DERIVATIVE.
%e A304323 The logarithmic derivative of A(x) yields the o.g.f. of A304313:
%e A304323 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 +...
%e A304323 INVERT TRANSFORM.
%e A304323 1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107675:
%e A304323 B(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + ... + A107675(n)*x^n + ...
%o A304323 (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]}
%o A304323 for(n=0,25, print1( a(n),", "))
%Y A304323 Cf. A304320, A304313, A304321, A304322, A304324, A304325.
%Y A304323 Cf. A006691, A107675.
%K A304323 nonn
%O A304323 0,3
%A A304323 _Paul D. Hanna_, May 11 2018