A360973 -id:A360973 - cp's OEIS frontend

A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(2*n) / n!.

1, 2, 18, 260, 4890, 110124, 2844772, 82196424, 2613699450, 90450874860, 3379153837180, 135445714293720, 5796441493971284, 263784018974675416, 12721572505160772840, 648250134428292640272, 34809708051186914034730, 1965040180185473309749788, 116359823755204505172646204

Offset: 0

Paul D. Hanna, Feb 27 2023

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + 2613699450*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^2) + (d^2/dx^2 x^4*A(x)^4)/2! + (d^3/dx^3 x^6*A(x)^6)/3! + (d^4/dx^4 x^8*A(x)^8)/4! + (d^5/dx^5 x^10*A(x)^10)/5! + (d^6/dx^6 x^12*A(x)^12)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^2), which begins
B(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ... + A360977(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^2 + (d/dx x^3*A(x)^4)/2! + (d^2/dx^2 x^5*A(x)^6)/3! + (d^3/dx^3 x^7*A(x)^8)/4! + (d^4/dx^4 x^9*A(x)^10)/5! + (d^5/dx^5 x^11*A(x)^12)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(2*n))/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..299

Cf. A360950, A360973, A360975, A360977.

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^2 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^2).
(3) B(x - x^2*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(2*n) / n! ) is the g.f. of A360977.
(4) a(n) = (n+1) * A360977(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/2)^n, where alfa = 2.498459235192... and c = 0.0920029178453... - Vaclav Kotesovec, Feb 28 2023
alfa = 5*LambertW(1/2) + 1/(1 + LambertW(1/2)). - Vaclav Kotesovec, Mar 13 2023

A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(3*n) / n!.

Original entry on oeis.org

1, 2, 24, 476, 12380, 386220, 13821276, 552876504, 24318017424, 1162989779660, 59987353249560, 3316841570302680, 195648523068917828, 12263065689662763024, 814027053454694421000, 57059908657536257254704, 4212606733712173668180012, 326799750176069289173027820

Offset: 0

Paul D. Hanna, Feb 27 2023

nonn

			G.f.: A(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + 24318017424*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^3) + (d^2/dx^2 x^4*A(x)^6)/2! + (d^3/dx^3 x^6*A(x)^9)/3! + (d^4/dx^4 x^8*A(x)^12)/4! + (d^5/dx^5 x^10*A(x)^15)/5! + (d^6/dx^6 x^12*A(x)^18)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^3), which begins
B(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ... + A360978(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^3 + (d/dx x^3*A(x)^6)/2! + (d^2/dx^2 x^5*A(x)^9)/3! + (d^3/dx^3 x^7*A(x)^12)/4! + (d^4/dx^4 x^9*A(x)^15)/5! + (d^5/dx^5 x^11*A(x)^18)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(3*n))/n! + ... ).

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

Cf. A360950, A360973, A360974, A360978.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(3*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^3 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^3).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(3*n) / n! ) is the g.f. of A360978.
(4) a(n) = (n+1) * A360978(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/3)^n, where alfa = 2.598541481443... and c = 0.058191982295165... - Vaclav Kotesovec, Feb 28 2023
alfa = 7*LambertW(1/3) + 1/(1 + LambertW(1/3)). - Vaclav Kotesovec, Mar 13 2023

A360976 G.f. satisfies: A(x) = Series_Reversion(x - x^3*A'(x)).

Original entry on oeis.org

1, 1, 6, 66, 1027, 20274, 479403, 13118703, 406181493, 14007078204, 531778565544, 22028404578840, 988535991793203, 47773653611710429, 2473958531200630992, 136684964338470273828, 8026375457238402039978, 499251236257852169668461, 32794618460003080060574283

Offset: 1

Paul D. Hanna, Feb 27 2023

nonn

a(n) = A360973(n-1)/(2*n-1) for n >= 1.

			G.f.: A(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)) = x, where
A'(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + ... + A360973(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)) + (d^2/dx^2 x^6*A'(x)^2)/2! + (d^3/dx^3 x^9*A'(x)^3)/3! + (d^4/dx^4 x^12*A'(x)^4)/4! + (d^5/dx^5 x^15*A'(x)^5)/5! + (d^6/dx^6 x^18*A'(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^2*A'(x) + (d/dx x^5*A'(x)^2)/2! + (d^2/dx^2 x^8*A'(x)^3)/3! + (d^3/dx^3 x^11*A'(x)^4)/4! + (d^4/dx^4 x^14*A'(x)^5)/5! + (d^5/dx^5 x^17*A'(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^n)/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A360973, A229619, A360977, A360978.

{a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*A'+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)).
(2) A(x) = x + A(x)^3 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^n / n!.

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(2*n) / n!.

Original entry on oeis.org

1, 3, 45, 1113, 36459, 1448568, 66726309, 3469988835, 200242815669, 12670449226269, 871389659249424, 64693985439491127, 5156607707368927875, 439261264283443326927, 39831856169938193953827, 3831650468281643037364389, 389807188331526942149375433

Offset: 0

Paul D. Hanna, Mar 03 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^2) + (d^2/dx^2 x^6*A(x)^4)/2! + (d^3/dx^3 x^9*A(x)^6)/3! + (d^4/dx^4 x^12*A(x)^8)/4! + (d^5/dx^5 x^15*A(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^2), which begins
B(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + 11778989157*x^17 + ... + A361047(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^2 + (d/dx x^5*A(x)^4)/2! + (d^2/dx^2 x^8*A(x)^6)/3! + (d^3/dx^3 x^11*A(x)^8)/4! + (d^4/dx^4 x^14*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(2*n))/n! + ... ).

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

Cf. A361047, A360950, A360973, A360974, A360975, A360976.

Cf. A229619, A356848, A088714, A303063.

nt = 40; (* number of terms to produce *)
A[_] = 0;
Do[A[x_] = D[InverseSeries[x - x^3*A[x]^2 + O[x]^k] // Normal, x], {k, 1, 2*nt}];
CoefficientList[A[x^(1/2)], x] (* Jean-François Alcover, Mar 04 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(2*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^2 +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^2).
(3) B(x - x^3*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361047.
(4) a(n) = (2*n+1) * A361047(n+1) for n >= 0.
a(n) == 0 (mod 3) for n > 0.
a(n) ~ c * 2^n * n! * n^alfa / LambertW(1/2)^n, where alfa = 1.623844426394406... and c = 0.18597481905555548924712403113114... - Vaclav Kotesovec, Mar 04 2023
alfa = (15*LambertW(1/2) - 1 + 3/(1 + LambertW(1/2)))/4. - Vaclav Kotesovec, Mar 15 2023

A361536 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(3*n) / n!.

Original entry on oeis.org

1, 3, 60, 2037, 92187, 5066952, 322801089, 23197971285, 1848188250810, 161297106209607, 15285968218925460, 1562519987561305566, 171348519312001997550, 20068058089211306151393, 2500498134501774994768119, 330350627790472265384885061, 46136067767500181432129130897

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^3) + (d^2/dx^2 x^6*A(x)^6)/2! + (d^3/dx^3 x^9*A(x)^9)/3! + (d^4/dx^4 x^12*A(x)^12)/4! + (d^5/dx^5 x^15*A(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^3), which begins
B(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + ... + A361302(n+1)*x^(2*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^3 + (d/dx x^5*A(x)^6)/2! + (d^2/dx^2 x^8*A(x)^9)/3! + (d^3/dx^3 x^11*A(x)^12)/4! + (d^4/dx^4 x^14*A(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(3*n))/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361537, A361541, A361542, A361543, A361551.

Cf. A361302.

nmax = 20; r = 3; s = 3; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(3*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^3 +O(x^(2*n+2))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^3).
(3) B(x - x^3*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(3*n) / n! ) is the g.f. of A361302.
(4) a(n) = (2*n+1) * A361302(n+1) for n >= 0.
a(n) ~ c * 2^n * n! * n^((21*LambertW(1/3) - 1 + 3/(1 + LambertW(1/3)))/4) / LambertW(1/3)^n, where c = 0.123530460429388663183565497... - Vaclav Kotesovec, Mar 16 2023

A361537 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(4*n) / n!.

Original entry on oeis.org

1, 3, 75, 3234, 186471, 13063908, 1060481214, 97053553710, 9840717984447, 1092337371705273, 131589391554509112, 17089208887923714204, 2379797411747290723350, 353790840030976298935989, 55935780589531899802966062, 9373903063348266793396858620

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^4) + (d^2/dx^2 x^6*A(x)^8)/2! + (d^3/dx^3 x^9*A(x)^12)/3! + (d^4/dx^4 x^12*A(x)^16)/4! + (d^5/dx^5 x^15*A(x)^20)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(4*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^4), which begins
B(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + ... + A361307(n+1)*x^(2*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^4 + (d/dx x^5*A(x)^8)/2! + (d^2/dx^2 x^8*A(x)^12)/3! + (d^3/dx^3 x^11*A(x)^16)/4! + (d^4/dx^4 x^14*A(x)^20)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(4*n))/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361541, A361542, A361543, A361551.

Cf. A361307.

nmax = 20; r = 3; s = 4; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(4*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^4 +O(x^(2*n+2))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(4*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^4).
(3) B(x - x^3*A(x)^4) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(4*n) / n! ) is the g.f. of A361307.
(4) a(n) = (2*n+1) * A361307(n+1) for n >= 0.
a(n) ~ c * 2^n * n! * n^((27*LambertW(1/4) - 1 + 3/(1 + LambertW(1/4)))/4) / LambertW(1/4)^n, where c = 0.09189708135429625612601629... - Vaclav Kotesovec, Mar 16 2023

A361541 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^n / n!.

Original entry on oeis.org

1, 4, 56, 1220, 34788, 1203152, 48418384, 2210163032, 112501779300, 6308565897088, 386149471644704, 25614932030415636, 1830512170952711968, 140224558208217547440, 11464991752291729651224, 996723500374559386157920, 91824970792933898453830680

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

			G.f.: A(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + a(n)*x^(3*n) + ...
where
A(x) = 1 + (d/dx x^4*A(x)) + (d^2/dx^2 x^8*A(x)^2)/2! + (d^3/dx^3 x^12*A(x)^3)/3! + (d^4/dx^4 x^16*A(x)^4)/4! + (d^5/dx^5 x^20*A(x)^5)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^4*A(x)), which begins
B(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + ... + A361308(n+1)*x^(3*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^3*A(x) + (d/dx x^7*A(x)^2)/2! + (d^2/dx^2 x^11*A(x)^3)/3! + (d^3/dx^3 x^15*A(x)^4)/4! + (d^4/dx^4 x^19*A(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^n)/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361542, A361543, A361551.

Cf. A361308.

nmax = 20; r = 4; s = 1; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(1*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^1 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^4*A(x)).
(3) B(x - x^4*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^n / n! ) is the g.f. of A361308.
(4) a(n) = (3*n+1) * A361308(n+1) for n >= 0.
a(n) ~ c * 3^n * n! * n^((6*LambertW(1) - 1 + 2/(1 + LambertW(1)))/3) / LambertW(1)^n, where c = 0.4150199735492722539427... - Vaclav Kotesovec, Mar 16 2023

A361542 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(2*n) / n!.

Original entry on oeis.org

1, 4, 84, 2940, 137228, 7809680, 517517212, 38860889496, 3248881861500, 298704250964336, 29928006672383280, 3244628959712243628, 378449007991303855532, 47261928190105905687600, 6293239981401396941576632, 890249832854933140207681360, 133355904852469516343820132852

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

			G.f.: A(x) = 1 + 4*x^3 + 84*x^6 + 2940*x^9 + 137228*x^12 + 7809680*x^15 + 517517212*x^18 + 38860889496*x^21 + ... + a(n)*x^(3*n) + ...
where
A(x) = 1 + (d/dx x^4*A(x)^2) + (d^2/dx^2 x^8*A(x)^4)/2! + (d^3/dx^3 x^12*A(x)^6)/3! + (d^4/dx^4 x^16*A(x)^8)/4! + (d^5/dx^5 x^20*A(x)^10)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^4*A(x)^2), which begins
B(x) = x + x^4 + 12*x^7 + 294*x^10 + 10556*x^13 + 488105*x^16 + 27237748*x^19 + ... + A361309(n+1)*x^(3*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^3*A(x)^2 + (d/dx x^7*A(x)^4)/2! + (d^2/dx^2 x^11*A(x)^6)/3! + (d^3/dx^3 x^15*A(x)^8)/4! + (d^4/dx^4 x^19*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^(2*n))/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361541, A361543, A361551.

Cf. A361309.

nmax = 20; r = 4; s = 2; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(2*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^2 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^4*A(x)^2).
(3) B(x - x^4*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361309.
(4) a(n) = (3*n+1) * A361309(n+1) for n >= 0.
a(n) ~ c * 3^n * n! * n^((10*LambertW(1/2) - 1 + 2/(1 + LambertW(1/2)))/3) / LambertW(1/2)^n, where c = 0.21905012601982153625576836... - Vaclav Kotesovec, Mar 16 2023

A361543 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(3*n) / n!.

Original entry on oeis.org

1, 4, 112, 5380, 346788, 27285968, 2498963752, 259124694312, 29885849525700, 3786931724896768, 522451837498888672, 77929657518224116484, 12496899169394954817144, 2144326582901160246138160, 392104633203721656029928184, 76134826269461672101153285664

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

			G.f.: A(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + a(n)*x^(3*n) + ...
where
A(x) = 1 + (d/dx x^4*A(x)^3) + (d^2/dx^2 x^8*A(x)^6)/2! + (d^3/dx^3 x^12*A(x)^9)/3! + (d^4/dx^4 x^16*A(x)^12)/4! + (d^5/dx^5 x^20*A(x)^15)/5! + ... + (d^n/dx^n x^(4*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^4*A(x)^3), which begins
B(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + ... + A361310(n+1)*x^(3*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^3*A(x)^3 + (d/dx x^7*A(x)^6)/2! + (d^2/dx^2 x^11*A(x)^9)/3! + (d^3/dx^3 x^15*A(x)^12)/4! + (d^4/dx^4 x^19*A(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A(x)^(3*n))/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361541, A361542, A361551.

Cf. A361310.

nmax = 20; r = 4; s = 3; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(4*m)*A^(3*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^4*A^3 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(3*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^4*A(x)^3).
(3) B(x - x^4*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A(x)^(3*n) / n! ) is the g.f. of A361310.
(4) a(n) = (3*n+1) * A361310(n+1) for n >= 0.
a(n) ~ c * 3^n * n! * n^((14*LambertW(1/3) - 1 + 2/(1 + LambertW(1/3)))/3) / LambertW(1/3)^n, where c = 0.147646967535758066931690294... - Vaclav Kotesovec, Mar 16 2023

A361551 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n (x^(5n) A(x)^n) / n!.

Original entry on oeis.org

1, 5, 90, 2535, 93840, 4226355, 222038775, 13259599965, 884588496165, 65114097133590, 5239173990133060, 457392343670390700, 43064135370809341250, 4350264113638902544555, 469422682906897831519170, 53897717818214315584719430, 6561919113715122121302125775

Offset: 0

Paul D. Hanna, Mar 15 2023

nonn

Conjecture: If r>=2 and s>=1 and A(x) = Sum_{n>=0} d^n/dx^n x^(r*n) * A(x)^(s*n) / n!, then a(n) ~ c(r,s) * n! * n^alpha(r,s) * ((r-1)/LambertW(1/s))^n, where alpha(r,s) = ((2*s+1)*LambertW(1/s) + 1 + 1/(1 + LambertW(1/s))) * r/(2*(r-1)) - 1 and c(r,s) is a constant independent on n. - Vaclav Kotesovec, Mar 16 2023

			G.f.: A(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + 13259599965*x^28 + ... + a(n)*x^(4*n) + ...
where
A(x) = 1 + (d/dx x^5*A(x)) + (d^2/dx^2 x^10*A(x)^2)/2! + (d^3/dx^3 x^15*A(x)^3)/3! + (d^4/dx^4 x^20*A(x)^4)/4! + (d^5/dx^5 x^25*A(x)^5)/5! + ... + (d^n/dx^n x^(5*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^5*A(x)), which begins
B(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + ... + A361311(n+1)*x^(4*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^4*A(x) + (d/dx x^9*A(x)^2)/2! + (d^2/dx^2 x^14*A(x)^3)/3! + (d^3/dx^3 x^19*A(x)^4)/4! + (d^4/dx^4 x^24*A(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A(x)^n)/n! + ... ).

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

Cf. A360950, A360974, A360975, A360973, A361046, A361536, A361537, A361541, A361542, A361543.

Cf. A361311.

nmax = 20; r = 5; s = 1; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(5*m)*A^(1*m)/m!)) +O(x^(4*n+1))); polcoeff(A, 4*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^5*A^1 +O(x^(4*n+2))))); polcoeff(A, 4*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(4*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^5*A(x)).
(3) B(x - x^5*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A(x)^n / n! ) is the g.f. of A361311.
(4) a(n) = (4*n+1) * A361311(n+1) for n >= 0.
a(n) ~ c * 4^n * n! * n^((15*LambertW(1) - 3 + 5/(1 + LambertW(1)))/8) / LambertW(1)^n, where c = 0.438413009716541247480505206... - Vaclav Kotesovec, Mar 16 2023

cp's OEIS Frontend

A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.

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A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.

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A360976 G.f. satisfies: A(x) = Series_Reversion(x - x^3*A'(x)).

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A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.

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A361536 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.

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A361537 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(4*n) / n!.

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A361541 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.

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A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(2*n) / n!.

A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(3*n) / n!.

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(2*n) / n!.

A361536 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(3*n) / n!.

A361537 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(4*n) / n!.

A361541 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^n / n!.

A361542 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(2*n) / n!.

A361543 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(3*n) / n!.

A361551 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n (x^(5n) A(x)^n) / n!.