A376176 - OEIS

%I #7 Sep 23 2024 08:10:40

%S 1,1,6,55,622,8015,113164,1711898,27357970,457507917,7952476482,

%T 142972019125,2648639456048,50415218306637,983728646223556,

%U 19641163430509505,400671660024507294,8340743906266061866,176998642509849677206,3825680705425292568049,84159282700462688412042

%N G.f. satisfies: x = A( x - A(x)^4/x^2 ).

%H Paul D. Hanna, <a href="/A376176/b376176.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) x = A( x - A(x)^4/x^2 ).

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

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

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

%e G.f.: A(x) = x + x^2 + 6*x^3 + 55*x^4 + 622*x^5 + 8015*x^6 + 113164*x^7 + 1711898*x^8 + 27357970*x^9 + 457507917*x^10 + ...

%e where x = A( x - A(x)^4/x^2 ).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 122*x^5 + 1390*x^6 + 17934*x^7 + 252847*x^8 + 3814724*x^9 + ...

%e A(x)^3 = x^3 + 3*x^4 + 21*x^5 + 202*x^6 + 2322*x^7 + 30030*x^8 + 423111*x^9 + 6369930*x^10 + ...

%e where A(x)^3 = x*A(x)^2 + A(A(x))^4.

%e A(x)^4 = x^4 + 4*x^5 + 30*x^6 + 296*x^7 + 3437*x^8 + 44600*x^9 + 628454*x^10 + 9446280*x^11 + ...

%e A(A(x))^4 = x^4 + 8*x^5 + 80*x^6 + 932*x^7 + 12096*x^8 + 170264*x^9 + 2555206*x^10 + 40413484*x^11 + ...

%e where A(x) = x + A(A(x))^4 / A(x)^2.

%e A(A(x)) = x + 2*x^2 + 14*x^3 + 141*x^4 + 1712*x^5 + 23392*x^6 + 347444*x^7 + 5498681*x^8 + 91552406*x^9 + ...

%e A(A(x))^2/A(x) = x + 3*x^2 + 23*x^3 + 242*x^4 + 3017*x^5 + 41965*x^6 + 631381*x^7 + 10089533*x^8 + 169256922*x^9 + ...

%o (PARI) {a(n) = my(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^4/x^2 +x*O(x^n))); polcoeff(A, n))}

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

%o (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}

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

%o (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m+1))/m!)+x*O(x^n))); polcoeff(A, n)}

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

%Y Cf. A213639.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 21 2024