A296170 -id:A296170 - cp's OEIS frontend

A296172 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3) for n>=1.

1, 1, -5, -197, -65111, -62390159, -125012786669, -447082993406405, -2583111044504384687, -22511408975342644804991, -281350305428215911326408789, -4850582201056517165575319399909, -111834955668396093904661955538037255, -3361788412998032560821833199260880942287, -128987969989211586699135087535153035663946301, -6203990036027464835833031041177436339788197962789

Offset: 0

Paul D. Hanna, Dec 07 2017

sign

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
To illustrate [x^(n-1)] A(x)^(n^3) = [x^n] A(x)^(n^3), form a table of coefficients of x^k in A(x)^(n^3) that begins as
n=1: [(1), (1), -5/2, -197/6, -65111/24, -62390159/120, -125012786669/720, ...];
n=2: [1, (8), (8), -1040/3, -71152/3, -64676744/15, -63817770776/45, ...];
n=3: [1, 27, (567/2), (567/2), -787941/8, -648507951/40, -405807483249/80, ...];
n=4: [1, 64, 1856, (88448/3), (88448/3), -689015872/15, -611019817664/45, ...];
n=5: [1, 125, 14875/2, 1649375/6, (156207625/24), (156207625/24), ...];
n=6: [1, 216, 22680, 1533168, 73812816, (12455715384/5), (12455715384/5), ...];
n=7: [1, 343, 115591/2, 38174185/6, 12294445009/24, 3808296195823/120, (1051338418817239/720), (1051338418817239/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^3) by n^3, we obtain the following sequence:
[1, 1, 21, 2764, 1249661, 1383968376, 3065126585473, 11913154589356672, 74286423963211939641, 696469981042645688972800, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...

Paul D. Hanna, Table of n, a(n) for n = 0..180

Cf. A296173, A296170, A296174, A296176.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

The logarithm of the e.g.f. A(x) is an integer series:
 log(A(x)) = Sum{n>=1} A296173(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^3) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^3) for n>=1, when evaluated at x = 0.
a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(3*n - 3) / (c^n * (3-c)^(2*n - 3) * exp(3*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A296174 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^4) = [x^n] A(x)^(n^4) for n>=1.

Original entry on oeis.org

1, 1, -13, -2999, -8197751, -81738176899, -2105524335759389, -115916378979693710123, -12069952631345502122877199, -2179911119857340269414590758951, -639738016495616440994202167765715629, -289812262583683385183617291938537580840159, -194420626455357631368336026954933981532680935943, -186615832949734453391125561079799823405868770406129579

Offset: 0

Paul D. Hanna, Dec 07 2017

sign

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! - 639738016495616440994202167765715629*x^10/10! +...
To illustrate [x^(n-1)] A(x)^(n^4) = [x^n] A(x)^(n^4), form a table of coefficients of x^k in A(x)^(n^4) that begins as
n=1: [(1), (1), -13/2, -2999/6, -8197751/24, -81738176899/120, ...];
n=2: [1, (16), (16), -26992/3, -16767472/3, -164706495728/15, ...];
n=3: [1, 81, (5427/2), (5427/2), -246155517/8, -2300014714833/40, ...];
n=4: [1, 256, 30976, (6633728/3), (6633728/3), -2939838787328/15, ...];
n=5: [1, 625, 381875/2, 225885625/6, (122571375625/24), (122571375625/24), ...];
n=6: [1, 1296, 830736, 350400816, 108698540976, (126219948303024/5), (126219948303024/5), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^4) by n^4, we obtain the following sequence:
[1, 1, 67, 51826, 196114201, 2337406450056, 68145136372652611, 4136219111307043556272, 467591060765602023501093201, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 - 176288642409787912257773903552*x^10 - 7260231964238768891891716773249396*x^11 - 405879958110794676900559524931590299892*x^12 +...

Paul D. Hanna, Table of n, a(n) for n = 0..160

Cf. A296175, A296170, A296172, A296176.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^4 ); n!*A[n+1]}
for(n=0,30,print1(a(n),", "))

The logarithm of the e.g.f. A(x) is an integer series:
 log(A(x)) = Sum{n>=1} A296175(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^4) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^4) for n>=1, when evaluated at x = 0.
a(n) ~ -sqrt(1-c) * 2^(8*n - 8) * n^(4*n - 4) / (c^n * (4-c)^(3*n - 4) * exp(4*n)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647386463779474372... - Vaclav Kotesovec, Oct 13 2020

A296176 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5) for n>=1.

Original entry on oeis.org

1, 1, -29, -36629, -734559239, -71200423546199, -22459270436075644469, -18407129959728493123679069, -33747438879000326056232288023439, -124162549312926509293620790889452447919, -843670934957017748849439817665935283173590349, -9914324850699841477684471316247032518786477385700389, -191047752973105011101288266443568575709649708408401069796759

Offset: 0

Paul D. Hanna, Dec 07 2017

sign

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x - 29*x^2/2! - 36629*x^3/3! - 734559239*x^4/4! - 71200423546199*x^5/5! - 22459270436075644469*x^6/6! - 18407129959728493123679069*x^7/7! - 33747438879000326056232288023439*x^8/8! - 124162549312926509293620790889452447919*x^9/9! - 843670934957017748849439817665935283173590349*x^10/10! +...
To illustrate [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5), form a table of coefficients of x^k in A(x)^(n^5) that begins as
n=1: [(1), (1), -29/2, -36629/6, -734559239/24, -71200423546199/120, ...];
n=2: [1, (32), (32), -614336/3, -2956631488/3, -285257147669696/15, ...];
n=3: [1, 243, (51759/2), (51759/2), -62010059733/8, -5840748850240719/40, ...];
n=4: [1, 1024, 508928, (470976512/3), (470976512/3), -9540780758505472/15, ...];
n=5: [1, 3125, 9671875/2, 29524484375/6, (86178242265625/24), (86178242265625/24), ...];
n=6: [1, 7776, 30116448, 77409815616, 148214160396864, (1099707612312815424/5), (1099707612312815424/5), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^5) by n^5, we obtain the following sequence:
[1, 1, 213, 919876, 27577037525, 3394159297261776, 1269158820664910885737, 1186717596374463676630699264, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 15*x^2 - 6090*x^3 - 30600650*x^4 - 593306350650*x^5 - 31192838317208826*x^6 - 3652177141294409632400*x^7 - 836986399841753367052602000*x^8 - 342157863774785896821739864893375*x^9 - 232492750600387706453977026534258393375*x^10 +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A296177, A296170, A296172, A296174.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^5 ); n!*A[n+1]}
for(n=0,30,print1(a(n),", "))

The logarithm of the e.g.f. A(x) is an integer series:
 log(A(x)) = Sum{n>=1} A296177(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^5) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^5) for n>=1, when evaluated at x = 0.

A300596 E.g.f. A(x) satisfies: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4) for n>=1.

Original entry on oeis.org

1, 1, 17, 13171, 56479849, 738706542221, 22885801082965201, 1448479282286023114807, 169382934361790242266135761, 33954915787325983176711221469529, 10997512067125948734754888814957997361, 5482894935903399886164748355296587003210971, 4041251688669102134446309448401146782811371078137

Offset: 0

Paul D. Hanna, Mar 09 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n^4) = n^3 * [x^(n-1)] exp(x)^(n^4) for n>=1.

			E.g.f.: A(x) = 1 + x + 17*x^2/2! + 13171*x^3/3! + 56479849*x^4/4! + 738706542221*x^5/5! + 22885801082965201*x^6/6! + 1448479282286023114807*x^7/7! + 169382934361790242266135761*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^(n^4) begins:
n=1: [(1), (1), 17/2, 13171/6, 56479849/24, 738706542221/120, ...];
n=2: [1, (16), (256), 113168/3, 114614528/3, 1486010366512/15, ...];
n=3: [1, 81, (7857/2), (636417/2), 1671341283/8, 20586397669407/40, ...];
n=4: [1, 256, 34816, (11641088/3), (2980118528/3), 26464517792512/15, ...];
n=5: [1, 625, 400625/2, 271091875/6, (232095075625/24), (145059422265625/24), ...];
n=6: [1, 1296, 850176, 379068336, 133027474176, (243163666719504/5), (315140112068477184/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 256 = 2^4*16; 636417/2 = 3^4*7857/2; 2980118528/3 = 4^4*11641088/3; ...
illustrating that: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + 31779658925496*x^6 + 287364845865893467*x^7 + 4200677982722915635200*x^8 + ... + A300597(n)*x^n + ...

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

Cf. A182962, A296170, A300590, A300592, A300594, A300597.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = ((#A-1)^4*V[#A-1] - V[#A])/(#A-1)^4 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300597(n)*x^n, a power series in x with integer coefficients.

A300614 E.g.f. A(x) satisfies: [x^n] A(x)^(n^5) = n^5 * [x^(n-1)] A(x)^(n^5) for n>=1.

Original entry on oeis.org

1, 1, 33, 118195, 3419881993, 412433022394701, 152749066271797582081, 143430189975946314906194983, 297572051428536567500380512047505, 1228369468294423956894049108209998483353, 9295358239339907973775754707697954813272247041, 120806095217585335844962641542342569940874366294995451

Offset: 0

Paul D. Hanna, Mar 10 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n^5) = n^4 * [x^(n-1)] exp(x)^(n^5) for n>=1.

			E.g.f.: A(x) = 1 + x + 33*x^2/2! + 118195*x^3/3! + 3419881993*x^4/4! + 412433022394701*x^5/5! + 152749066271797582081*x^6/6! + 143430189975946314906194983*x^7/7! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^(n^5) begins:
n=1: [(1), (1), 33/2, 118195/6, 3419881993/24, 137477674131567/40, ...];
n=2: [1, (32), (1024), 1955104/3, 13739402240/3, 1651861749195104/15, ...];
n=3: [1, 243, (66825/2), (16238475/2), 288411062643/8, 33749327928610701/40, ...];
n=4: [1, 1024, 540672, (647668736/3), (663212785664/3), 18460138990560256/5, ...];
n=5: [1, 3125, 9865625/2, 31824134375/6, (116555654565625/24), (364236420517578125/24), ...];
n=6: [1, 7776, 30357504, 79484677920, 158407197944832, (1433574291388125024/5), (11147473689834060186624/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 1024 = 2^5*32; 16238475/2 = 3^5*66825/2; 663212785664/3 = 4^5*647668736/3; ...
illustrating that: [x^n] A(x)^(n^5) = n^5 * [x^(n-1)] A(x)^(n^5).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 16*x^2 + 19683*x^3 + 142475264*x^4 + 3436799053125*x^5 + 212148041589128016*x^6 + 28458158819417861315152*x^7 + 7380230750280159370894934016*x^8 + ... + A300615(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300615.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = ((#A-1)^5*V[#A-1] - V[#A])/(#A-1)^5 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300615(n)*x^n, a power series in x with integer coefficients.

A300616 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n for n>=1.

Original entry on oeis.org

1, 1, 7, 199, 14065, 1924201, 445859911, 161145717727, 85790577700129, 64427620614173425, 65943035132156264071, 89425725156530626400791, 156922032757769223085752337, 349233620942232034199096926489, 968890106809715834110637461124935, 3301188169350221687517822373590448111, 13634136452997022097853039839798901714241

Offset: 0

Paul D. Hanna, Mar 10 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^n = [x^(n-1)] exp(x)^n for n>=1.

Compare to e.g.f. G(x) of A182962: [x^n] G(x)^n = n * [x^(n-1)] G(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14065*x^4/4! + 1924201*x^5/5! + 445859911*x^6/6! + 161145717727*x^7/7! + 85790577700129*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 7/2, 199/6, 14065/24, 1924201/120, 445859911/720, ...];
n=2: [1, (2), (8), 220/3, 3752/3, 502114/15, 57409744/45, ...];
n=3: [1, 3, (27/2), (243/2), 16035/8, 2098161/40, 157765131/80, ...];
n=4: [1, 4, 20, (536/3), (8576/3), 1096868/15, 121987336/45, ...];
n=5: [1, 5, 55/2, 1475/6, (91825/24), (2295625/24), 503279435/144, ...];
n=6: [1, 6, 36, 324, 4920, (601074/5), (21638664/5), 7491519768/35...];
n=7: [1, 7, 91/2, 2485/6, 147721/24, 17641687/120, (3752979139/720), (183895977811/720), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 8 = 2^2*2; 243/2 = 3^2*27/2; 8576/3 = 4^2*536/3; ...
illustrating that: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + 2093655600*x^8 + 175312873125*x^9 + 17987972309725*x^10 + ... + A300617(n)*x^n + ...

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

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300614, A300617.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300617(n)*x^n, a power series in x with integer coefficients.

a(n) ~ c * (n!)^3, where c = 1.685041722777551007711429045295022018562828... - Vaclav Kotesovec, Mar 10 2018

A300873 E.g.f. A(x) satisfies: [x^n] A(x)^(n(n+1)) = 2n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

Original entry on oeis.org

1, 1, 3, 43, 2041, 197721, 31094251, 7086479443, 2187876597873, 874871971357681, 438740658523346131, 269314248304239932091, 198529013874402868930153, 173067121551267519897494473, 176154202119865662835343738811, 207099741506845262022248534098531, 278645958801870115911315221474653921, 425605862347493892454320041743878801633

Offset: 0

Paul D. Hanna, Mar 14 2018

nonn

Compare to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
illustrating that: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ... + A300874(n)*x^n + ...

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

Cf. A300874, A300870, A295811, A300590, A296170, A182962.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); EGF=Ser(A); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * n!^2 / n^3, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.75891265... - Vaclav Kotesovec, Aug 11 2021

A300618 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n for n>=1.

Original entry on oeis.org

1, 1, 15, 1285, 347065, 224232501, 296201195791, 719274160258585, 2967337954539761265, 19563048191912257746505, 196302561889372679184550831, 2881342883089548932078551914861, 59862434550069057805236434063104105, 1712289828911477479390772271103153886845

Offset: 0

Paul D. Hanna, Mar 10 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^n = [x^(n-1)] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x + 15*x^2/2! + 1285*x^3/3! + 347065*x^4/4! + 224232501*x^5/5! + 296201195791*x^6/6! + 719274160258585*x^7/7! + 2967337954539761265*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 15/2, 1285/6, 347065/24, 74744167/40, ...];
n=2: [1, (2), (16), 1330/3, 88220/3, 56540144/15, ...];
n=3: [1, 3, (51/2), (1377/2), 358875/8, 228121101/40, ...];
n=4: [1, 4, 36, (2852/3), (182528/3), 38352496/5, ...];
n=5: [1, 5, 95/2, 7385/6, (1857145/24), (232143125/24), ...];
n=6: [1, 6, 60, 1530, 94500, (58551624/5), (12647150784/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 16 = 2^3*2; 1377/2 = 3^3*51/2; 182528/3 = 4^3*2852/3; ...
illustrating that: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + ... + A300619(n)*x^n + ...

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

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300614, A300616, A300619.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300619(n)*x^n, a power series in x with integer coefficients.

cp's OEIS Frontend

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A296176 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5) for n>=1.

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A300596 E.g.f. A(x) satisfies: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4) for n>=1.

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A300614 E.g.f. A(x) satisfies: [x^n] A(x)^(n^5) = n^5 * [x^(n-1)] A(x)^(n^5) for n>=1.

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A300616 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n for n>=1.

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A300873 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

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A300618 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n for n>=1.

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