A361046 -id:A361046 - cp's OEIS frontend

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

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

A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

Original entry on oeis.org

1, 1, 9, 159, 4051, 131688, 5132793, 231332589, 11778989157, 666865748751, 41494745678544, 2812781975630049, 206264308294757115, 16268935714201604701, 1373512281722006688063, 123601628009085259269819, 11812339040349301277253801, 1194940136210629914238593762

Offset: 1

Paul D. Hanna, Mar 03 2023

nonn

Conjecture: a(n) == 1 (mod 3) iff n = (3^k - 1)/2 for k >= 0, otherwise a(n) == 0 (mod 3).

			G.f.: A(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^2) = x, where
A'(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + A361046(n)*x^(2*n) + ...
Also,
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! + ...
Further,
A(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 = 1..301

Cf. A361046, A229619, A360977, A360978.

{a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*(A')^2 +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).
(2) A(x) = x + A(x)^3 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(2*n) / n!, where A'(x) is the g.f. of A361046.
a(n) = A361046(n-1)/(2*n-1) for n >= 1.

A361303 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(3*n) / n!.

Original entry on oeis.org

1, 2, 15, 92, 615, 4200, 29190, 205416, 1458909, 10436030, 75079719, 542669244, 3937604853, 28664996080, 209261546580, 1531373181120, 11230365782130, 82512324300222, 607246350958449, 4475646134515360, 33031356134381220, 244073892799489500, 1805479496422561740

Offset: 0

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = 1 + 2*x + 15*x^2 + 92*x^3 + 615*x^4 + 4200*x^5 + 29190*x^6 + 205416*x^7 + 1458909*x^8 + 10436030*x^9 + ...

Cf. A361305, A215128, A361046.

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

/* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^3 )); 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) * (1 + x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + 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) * (1+x)^(3*n) / n! ) is the g.f. of A361305.
(4) a(n) = (n+1) * A361305(n+1) for n >= 0.

cp's OEIS Frontend

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

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

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

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A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

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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!.

A361303 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(3*n) / n!.