A361310 - OEIS

%I #9 Mar 19 2023 21:43:26

%S 1,1,16,538,26676,1705373,131524408,11778395196,1195433981028,

%T 135247561603456,16853285080609312,2292048750536003426,

%U 337754031605269049112,53608164572529006153454,9118712400086550140230888,1655104918901340697851158384,319341008921919836189242604080

%N G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^3).

%H Paul D. Hanna, <a href="/A361310/b361310.txt">Table of n, a(n) for n = 1..200</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.

%F (1) A(x) = Series_Reversion(x - x^4*A'(x)^3).

%F (2) A(x) = x + A(x)^4 * A'(A(x))^3.

%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(3*n) / n! ).

%F (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(3*n) / n! is the g.f. of A361543.

%F (5) a(n) = A361543(n-1)/(3*n-2) for n >= 1.

%e G.f.: A(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + 11778395196*x^22 + ... + a(n)*x^(3*n-2) + ...

%e By definition, A(x - x^4*A'(x)^3) = x, where

%e A'(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + A361543(n)*x^(3*n) + ...

%e Also,

%e 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! + ...

%e Further,

%e A(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! + ... ).

%o (PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^3 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A361543.

%Y Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361311.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 17 2023