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!.

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

Offset

0,2

Links

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

Formula

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

Example

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! + ... ).

Mathematica

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 *)

Prog

(PARI) {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), ", "))
(PARI) /* 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), ", "))

Crossrefs

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

Cf. A361307.

Sequence in context: A003690 A230143 A228841 * A249938 A195263 A034940

Adjacent sequences: A361534 A361535 A361536 * A361538 A361539 A361540

Keyword

nonn

Author

Paul D. Hanna, Mar 15 2023

Status

approved