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!.

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

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^(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

Example

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! + ... ).

Mathematica

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 *)

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^(4*m)*A^(2*m)/m!)) +O(x^(3*n+1))); polcoeff(A, 3*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^4*A^2 +O(x^(3*n+2))))); polcoeff(A, 3*n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

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

Cf. A361309.

Sequence in context: A156477 A024259 A357672 * A024260 A099706 A012065

Adjacent sequences: A361539 A361540 A361541 * A361543 A361544 A361545

Keyword

nonn

Author

Paul D. Hanna, Mar 15 2023

Status

approved