A304312 -id:A304312 - cp's OEIS frontend

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

Paul D. Hanna, May 11 2018

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.

			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.

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

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

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 *)

{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),", "));)

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

A304322 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 5, 54, 935, 22417, 685592, 25431764, 1106630687, 55174867339, 3097872254493, 193283918695494, 13260815963831108, 991928912663646012, 80325879518096889760, 7000127337189146831092, 653156403671376068448047, 64963788042207845593775999, 6861040250464949653809027311, 766815367797924824316405828466, 90417908118862070187113849296815

Offset: 0

Paul D. Hanna, May 11 2018

nonn

It is conjectured that the coefficients of o.g.f. A(x) consist entirely of integers.
Equals row 2 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A107668.
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 2 inputs (A006691).

			O.g.f.: A(x) = 1 + x + 5*x^2 + 54*x^3 + 935*x^4 + 22417*x^5 + 685592*x^6 + 25431764*x^7 + 1106630687*x^8 + 55174867339*x^9 + 3097872254493*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^2*x) / A(x) begins:
n=0: [1, -1, -8, -270, -19584, -2427000, -455544000, -120136161600, ...];
n=1: [1, 0, -9, -296, -20715, -2527704, -470405285, -123376631664, ...];
n=2: [1, 3, 0, -350, -24672, -2867256, -518870528, -133753337280, ...];
n=3: [1, 8, 55, 0, -29547, -3559056, -614943333, -153534305160, ...];
n=4: [1, 15, 216, 2674, 0, -4291704, -783235520, -187656684864, ...];
n=5: [1, 24, 567, 12880, 251541, 0, -948897125, -243358236600, ...];
n=6: [1, 35, 1216, 41634, 1372320, 38884296, 0, -295870371264, ...];
n=7: [1, 48, 2295, 109000, 5106453, 230531544, 8944955227, 0, ...];
n=8: [1, 63, 3960, 248050, 15443328, 949131144, 56257429312, 2865412167360, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304312:
A'(x)/A(x) = 1 + 9*x + 148*x^2 + 3493*x^3 + 106431*x^4 + 3950832*x^5 + 172325014*x^6 + 8617033285*x^7 + 485267003023*x^8 + 30363691715629*x^9 + ... + A304312(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A107668:
B(x) = 1 + 4*x + 45*x^2 + 816*x^3 + 20225*x^4 + 632700*x^5 + 23836540*x^6 + 1048592640*x^7 + 52696514169*x^8 + ... + A107668(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A304320, A304312, A304321, A304323, A304324, A304325.

Cf. A006691, A107668.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^2 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

a(n) ~ sqrt(1-c) * 2^(2*n - 1/2) * n^(n - 1/2) / (sqrt(Pi) * c^n * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A006691 Normalized number of connected (n+1)-state finite automata with 2 inputs.

Original entry on oeis.org

9, 148, 3493, 106431, 3950832, 172325014, 8617033285, 485267003023, 30363691715629, 2088698040637242, 156612539215405732, 12709745319947141220, 1109746209390479579732, 103724343230007402591558, 10332348604630683943445797, 1092720669631704348689818959, 122274820828415241343176467043

Offset: 1

N. J. A. Sloane

nonn

Is this sequence essentially the same as A304312? - Paul D. Hanna, May 11 2018
From Petros Hadjicostas, Feb 26 2021: (Start)
See Table 2 (p. 683) in Robinson (1984) for values of S(p)/(p-1)! = S(p,d)/(p-1)! with p >= 2 and d = 2. In the paper, S(p) = S(p,d) is the number of (labeled) strongly connected finite automata with state set {1, 2, ..., p} and d inputs (p. 680). Since the offset here is 1, the original name of the sequence was changed to read "(n+1)-state" from "n-state".
This change agrees with Valery A. Liskovets's formula below, who was the first one to derive expressions for the quantity S(p) = S(p,d) for a general d more than a decade before Robinson (1984). See Liskovets (1971), where S(p) = S(p,d), with d inputs, is denoted by sigma_r(n) with r = d (inputs) and n = p (number of states). For d = 2, the values of S(p) = S(p,d=2) = (p-1)!*a(p-1) for p >= 1 (with a(0) := 1) are given in A027834, which has the correct name.
We may suggest two possible names for a(n): (i) the normalized number of labeled strongly connected (n+1)-state finite automata with 2 inputs, or (ii) the number of unlabeled strongly connected (n+1)-state finite automata with 2 inputs and a starting gate. (For purely unlabeled strongly connected n-state finite automata with 2 inputs, see A027835, whose terms are calculated based on Valery A. Liskovets' formulas.) (End)

Robert W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651 (86g:05026).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..200
Valery A. Liskovets [ Liskovec ], Enumeration of nonisomorphic strongly connected automata, (in Russian); Vesti Akad. Nauk. Belarus. SSR, Ser. Phys.-Mat., No. 3, 1971, pp. 26-30, esp. p. 30 (Math. Rev. 46 #5081; Zentralblatt 224 #94053).
Valery A. Liskovets [ Liskovec ], A general enumeration scheme for labeled graphs, (in Russian); Dokl. Akad. Nauk. Belarus. SSR, Vol. 21, No. 6 (1977), pp. 496-499 (Math. Rev. 58 #21797; Zentralblatt 412 #05052).
R. W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651. (86g:05026). [Annotated scanned copy, with permission of the author.]

Cf. A027834, A304312.

v[r_, n_] := v[r, n] = If[n == 0, 1, n^(r*n) - Sum[Binomial[n, t] * n^(r*(n - t)) * v[r, t] , {t, 1, n - 1}]];
s[r_, n_] := s[r, n] = v[r, n] + Sum[Binomial[n - 1, t - 1] * v[r, n - t] * s[r, t], {t, 1, n - 1}]
A027834[n_] := s[2, n];
a[n_] := A027834[n + 1]/n!;
Array[a, 28] (* Jean-François Alcover, Aug 27 2019 *)

a(n) = A027834(n+1)/n!. - Valery A. Liskovets, May 21 2018

Extended using the formula by Valery A. Liskovets by Hugo Pfoertner, May 21 2018

Name edited by Petros Hadjicostas, Feb 26 2021 to agree with Robinson's and Liskovets' papers.

A304313 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

1, 49, 6877, 1854545, 807478656, 514798204147, 451182323794896, 519961864703259753, 762210147961330421167, 1384945048774500147047194, 3055115321627096660341307614, 8043516699726480852467167758419, 24915939138210507189761922944830006, 89709850983809128394441772076036629240, 371523831948166269091257380175120352465872

Offset: 0

Paul D. Hanna, May 11 2018

nonn

Is this sequence essentially the same as A006692?
Conjecture: a(n) is the number of connected n-state finite automata with 3 inputs (A006692).
Equals row 3 of table A304321.

			O.g.f.: L(x) = 1 + 49*x + 6877*x^2 + 1854545*x^3 + 807478656*x^4 + 514798204147*x^5 + 451182323794896*x^6 + 519961864703259753*x^7 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304323 :
F(x) = 1 + x + 25*x^2 + 2317*x^3 + 466241*x^4 + 162016980*x^5 + 85975473871*x^6 + 64545532370208*x^7 + 65062315637060121*x^8 + ... + A304323(n)*x^n + ...
which satisfies [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

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

Cf. A304323, A006692, A304321, A304312, A304314, A304315.

m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^3*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)

{a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^3 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))

Logarithmic derivative of the o.g.f. of A304323.
For n>=1, a(n) = B_{n+1}((n+1)^3-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - Max Alekseyev, Jun 18 2018
a(n) ~ sqrt(1-c) * 3^(3*(n+1)) * n^(2*n + 5/2) / (sqrt(2*Pi) * c^(n+1) * (3-c)^(2*(n+1)) * exp(2*n)), where c = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Aug 31 2020

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 11 2018

nonn

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).

			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 + ...

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

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

Cf. A006691, A107675.

{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),", "))

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

A304314 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

1, 225, 229000, 612243125, 3367384031526, 33056423981177346, 527146092112494861420, 12764850938355048224394925, 446065249480005516657138106375, 21615893741029073481369412949207860, 1406758471936562034421316174257309550136, 119755662436589797897149020637183902177930534

Offset: 0

Paul D. Hanna, May 11 2018

nonn

Conjecture: a(n) is the number of connected n-state finite automata with 4 inputs.

Equals row 4 of table A304321.

			O.g.f.: L(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
F(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ... + A304324(n)*x^n + ...
which satisfies [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A304324, A304321, A304312, A304313, A304315.

m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^4*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)

{a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))

Logarithmic derivative of the o.g.f. of A304324.
For n>=1, a(n) = B_{n+1}((n+1)^4-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - Max Alekseyev, Jun 18 2018
a(n) ~ sqrt(1-c) * 4^(4*(n+1)) * n^(3*n + 7/2) / (sqrt(2*Pi) * c^(n+1) * (4-c)^(3*(n+1)) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - Vaclav Kotesovec, Aug 31 2020

A304315 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

1, 961, 6737401, 172342090401, 11657788116175751, 1722786509653595220757, 489506033977061086758261063, 243968979437942649897623460813009, 199025593654123221838381793032781035510, 251774439716905627952289102887999425054599511, 472942802381336010263584088374665504251010554412128, 1273071332950625956697135571575613091625334028239417955701

Offset: 0

Paul D. Hanna, May 11 2018

nonn

Conjecture: a(n) is the number of connected n-state finite automata with 5 inputs.

Equals row 5 of table A304321.

			O.g.f.: L(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304325 :
F(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ... + A304325(n)*x^n + ...
which satisfies [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A304325, A304321, A304312, A304313, A304314.

m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n^5*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)

{a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^4 +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))

Logarithmic derivative of the o.g.f. of A304325.
For n>=1, a(n) = B_{n+1}((n+1)^5-0!*a(0),-1!*a(1),...,-(n-1)!*a(n-1),0) / n!, where B_{n+1}(...) is the (n+1)-st complete exponential Bell polynomial. - Max Alekseyev, Jun 18 2018
a(n) ~ sqrt(1-c) * 5^(5*(n+1)) * n^(4*n + 9/2) / (sqrt(2*Pi) * c^(n+1) * (5-c)^(4*(n+1)) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - Vaclav Kotesovec, Aug 31 2020

cp's OEIS Frontend

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.

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A304322 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x ) / A(x) = 0 for n>0.

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A006691 Normalized number of connected (n+1)-state finite automata with 2 inputs.

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A304313 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^3 * x ) / F(x) = 0 for n>0.

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

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A304314 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^4 * x ) / F(x) = 0 for n>0.

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A304315 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n^5 * x ) / F(x) = 0 for n>0.

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