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

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

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)^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

Example

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! + ... ).

Mathematica

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 *)

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^(1*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^1 +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, A361542, A361543, A361551.

Cf. A361308.

Sequence in context: A371678 A009058 A276363 * A218309 A061924 A183612

Adjacent sequences: A361538 A361539 A361540 * A361542 A361543 A361544

Keyword

nonn

Author

Paul D. Hanna, Mar 15 2023

Status

approved