A376176 G.f. satisfies: x = A( x - A(x)^4/x^2 ).

1, 1, 6, 55, 622, 8015, 113164, 1711898, 27357970, 457507917, 7952476482, 142972019125, 2648639456048, 50415218306637, 983728646223556, 19641163430509505, 400671660024507294, 8340743906266061866, 176998642509849677206, 3825680705425292568049, 84159282700462688412042

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

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

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

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

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

Example

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 + ...
where x = A( x - A(x)^4/x^2 ).
RELATED SERIES.
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 + ...
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 + ...
where A(x)^3 = x*A(x)^2 + A(A(x))^4.
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 + ...
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 + ...
where A(x) = x + A(A(x))^4 / A(x)^2.
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 + ...
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 + ...

Prog

(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))}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A213639.

Sequence in context: A372210 A365840 A006200 * A243691 A151345 A281596

Adjacent sequences: A376173 A376174 A376175 * A376177 A376178 A376179

Keyword

nonn

Author

Paul D. Hanna, Sep 21 2024

Status

approved