A361309 - OEIS

%I #11 Mar 19 2023 21:41:02

%S 1,1,12,294,10556,488105,27237748,1766404068,129955274460,

%T 10668008963012,965419570076880,95430263520948342,

%U 10228351567332536636,1181548204752647642190,146354418172125510269224,19353257235976807395819160,2721549078621826864159594548

%N G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^2).

%H Paul D. Hanna, <a href="/A361309/b361309.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)^2).

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

%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(2*n) / n! ).

%F (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(2*n) / n! is the g.f. of A361542.

%F (5) a(n) = A361542(n-1)/(3*n-2) for n >= 1.

%e G.f.: A(x) = x + x^4 + 12*x^7 + 294*x^10 + 10556*x^13 + 488105*x^16 + 27237748*x^19 + 1766404068*x^22 + 129955274460*x^25 + ... + a(n)*x^(3*n-2) + ...

%e By definition, A(x - x^4*A'(x)^2) = x, where

%e A'(x) = 1 + 4*x^3 + 84*x^6 + 2940*x^9 + 137228*x^12 + 7809680*x^15 + 517517212*x^18 + 38860889496*x^21 + ... + A361542(n)*x^(3*n) + ...

%e Also,

%e 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! + ...

%e Further,

%e A(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! + ... ).

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

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

%Y Cf. A361542.

%Y Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361310, A361311.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 17 2023