A304320 -id:A304320 - cp's OEIS frontend

A337551 Diagonal of A304320.

1, 1, 5, 2317, 153143499, 2331601789103231, 13956629370284243750857868, 50423205212193924706520234988903231406, 154976268837151170420647979803561273009994277188269267, 540838714291102105296873678266047207577891369813629960678503735540658879

Offset: 0

Vaclav Kotesovec, Aug 31 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..26

Cf. A304320.

{T(n, k) = my(A=[1], m); for(i=1, k, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^n +x*O(x^m)) / Ser(A) )[m] ); A[k+1]}
for(n=0, 10, print1(T(n, n), ", ")) \\ after Paul D. Hanna

a(n) ~ n^(n^2) / n!.

a(n) ~ exp(n) * n^(n^2 - n - 1/2) / sqrt(2*Pi).

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.

Original entry on oeis.org

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

A304319 O.g.f. A(x) satisfies: [x^n] exp( n(n+1) x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 2, 10, 104, 1772, 42408, 1303504, 48736000, 2139552016, 107629121888, 6094743943584, 383305860004992, 26491391713168640, 1994924925169038848, 162537118868301414912, 14243360542620058589184, 1335710880923054761115904, 133461369304858515494530560, 14154134380237986764584033792, 1587931951984022880659170662400

Offset: 0

Paul D. Hanna, May 11 2018

nonn

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

Note that if [x^n] exp( (n+1)*(n+2)*x ) / G(x) = 0 then G(x) does not consist entirely of integer coefficients.

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:
n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];
n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];
n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];
n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];
n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];
n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];
n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];
n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];
n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304317:
A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304317(n)*x^n + ...

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

Cf. A304317, A304318, A304320, A304863, A304864, A304865.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +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) * n^(n - 1/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304324 O.g.f. A(x) satisfies: [x^n] exp( n^4 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 113, 76446, 153143499, 673638499100, 5510097691767062, 75312181798660695788, 1595682359653020033714019, 49564410138113345565513815041, 2161639124039437373346491749452440, 127889301139607880711208251726358504898, 9979766671875039854419652569806336108694074

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 4 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304394.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304314.
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 4 inputs.

			O.g.f.: A(x) = 1 + x + 113*x^2 + 76446*x^3 + 153143499*x^4 + 673638499100*x^5 + 5510097691767062*x^6 + 75312181798660695788*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^4*x) / A(x) begins:
n=0: [1, -1, -224, -457326, -3671476224, -80797824300000, ...];
n=1: [1, 0, -225, -458000, -3673306875, -80816186256624, ...];
n=2: [1, 15, 0, -464750, -3701040000, -81092721606624, ...];
n=3: [1, 80, 6175, 0, -3787546875, -82312696206624, ...];
n=4: [1, 255, 64800, 15951250, 0, -84756571206624, ...];
n=5: [1, 624, 389151, 242091424, 146271536901, 0, ...];
n=6: [1, 1295, 1676800, 2170415250, 2804103120000, 3524906587193376, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^4*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304314:
A'(x)/A(x) = 1 + 225*x + 229000*x^2 + 612243125*x^3 + 3367384031526*x^4 + 33056423981177346*x^5 + 527146092112494861420*x^6 + ... + A304314(n)*x^n + ...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304394:
B(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ... + A304394(n)*x^n + ...

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

Cf. A304320, A304314, A304321, A304322, A304323, A304325.

Cf. A304394.

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

a(n) ~ sqrt(1-c) * 4^(4*n) * n^(3*n - 1/2) / (sqrt(2*Pi) * c^n * (4-c)^(3*n) * exp(3*n)), where c = -LambertW(-4*exp(-4)). - 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

A304325 O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 1, 481, 2246281, 43087884081, 2331601789103231, 287133439746933073357, 69929721774643572422651223, 30496192503451926066104677123329, 22113985380962062942048847693898939310, 25177466100486219354624677349405490885006591, 42994825404638061265611776726882581676486680632128

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 5 of table A304320.
O.g.f. A(x) = 1/(1 - x*B(x)), where B(x) is the o.g.f. of A304395.
Logarithmic derivative of o.g.f. A(x), A'(x)/A(x), equals o.g.f. of A304315.
Conjecture: given o.g.f. A(x), the coefficient of x^n in A'(x)/A(x) enumerates the connected n-state finite automata with 5 inputs.

			O.g.f.: A(x) = 1 + x + 481*x^2 + 2246281*x^3 + 43087884081*x^4 + 2331601789103231*x^5 + 287133439746933073357*x^6 + 69929721774643572422651223*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^5*x) / A(x) begins:
n=0: [1, -1, -960, -13471920, -1033995878400, -279781615181250000, ...];
n=1: [1, 0, -961, -13474802, -1034049771843, -279786785295370804, ...];
n=2: [1, 31, 0, -13534384, -1035725264896, -279947192760516048, ...];
n=3: [1, 242, 57603, 0, -1044001318107, -281045183102366562, ...];
n=4: [1, 1023, 1045568, 1054175056, 0, -284106842971323856, ...];
n=5: [1, 3124, 9758415, 30465809330, 93986716449725, 0, ...];
n=6: [1, 7775, 60449664, 469967719248, 3652476388472832, 28079364132086235696, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^5*x ) / A(x) = 0 for n >= 0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304315:
A'(x)/A(x) = 1 + 961*x + 6737401*x^2 + 172342090401*x^3 + 11657788116175751*x^4 + 1722786509653595220757*x^5 + 489506033977061086758261063*x^6 + ... + A304315(n)*x^n +...
INVERT TRANSFORM.
1/A(x) = 1 - x*B(x), where B(x) is the o.g.f. of A304395:
B(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + ... + A304395(n)*x^n + ...

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

Cf. A304320, A304315, A304321, A304322, A304323, A304324.

Cf. A304395.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^5 +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

a(n) ~ sqrt(1-c) * 5^(5*n) * n^(4*n - 1/2) / (sqrt(2*Pi) * c^n * (5-c)^(4*n) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - Vaclav Kotesovec, Aug 31 2020

A304318 O.g.f. A(x) satisfies: [x^n] exp( n(n-1) x ) / A(x) = 0.

Original entry on oeis.org

1, 0, 2, 24, 436, 10656, 328112, 12183456, 529242224, 26309617536, 1472135847072, 91526938123008, 6258004268952064, 466599240364076544, 37672137946943244288, 3274012281487011586560, 304724394621209905647360, 30239686358027369113804800, 3187164738879981461171955200, 355548230503664593634743375872

Offset: 0

Paul D. Hanna, May 11 2018

nonn

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

			O.g.f.: A(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n-1)*x) / A(x) begins:
n=0: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=1: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=2: [1, 2, 0, -160, -11600, -1376928, -250428416, -64479262720, ...];
n=3: [1, 6, 32, 0, -13392, -1630944, -286447104, -71981250048, ...];
n=4: [1, 12, 140, 1440, 0, -1916928, -351444096, -85338800640, ...];
n=5: [1, 20, 396, 7616, 128512, 0, -417488000, -107269127680, ...];
n=6: [1, 30, 896, 26496, 760752, 19101600, 0, -128348167680, ...];
n=7: [1, 42, 1760, 73440, 3034800, 121743072, 4260708864, 0, ...];
n=8: [1, 56, 3132, 174800, 9716608, 535021056, 28597069696, 1331047703552, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304316:
A'(x)/A(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ... + A304316(n)*x^n + ...

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

Cf. A304316, A304319, A304320.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m-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) * n^(n - 1/2) / (sqrt(Pi) * c^(n - 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

cp's OEIS Frontend

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A304318 O.g.f. A(x) satisfies: [x^n] exp( n*(n-1) * x ) / A(x) = 0.

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A304318 O.g.f. A(x) satisfies: [x^n] exp( n(n-1) x ) / A(x) = 0.