A302705 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.

1, 1, 3, 11, 43, 171, 677, 2637, 10035, 37171, 134009, 472785, 1655845, 5910373, 22254507, 90625475, 396822579, 1803795507, 8151776201, 35314777505, 142395796689, 518352934225, 1625522953935, 3944383216263, 4604242439037, -17114536692099, -114353748666873, -52384917067153, 4112292989447275, 42810794269242411, 290607272326013813

Offset: 0

Author

Paul D. Hanna, Apr 15 2018

Keywords

• sign

Comments

Note: if F(x) = 1 + Integral (x*F(x)^3)' / (x*F(x))' dx, then F(x) does not consist entirely of integer coefficients in its power series expansion.

Given G(x) = 1 + Integral (x*G(x)^p)' / (x*G(x)^q)' dx, for what fixed integers p and q is G(x) an integer power series?

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 171*x^5 + 677*x^6 + 2637*x^7 + 10035*x^8 + 37171*x^9 + 134009*x^10 + 472785*x^11 + 1655845*x^12 + ...
RELATED SERIES.
(x*A(x)^4)' / (x*A(x))' = 1 + 6*x + 33*x^2 + 172*x^3 + 855*x^4 + 4062*x^5 + 18459*x^6 + 80280*x^7 + 334539*x^8 + ... + (n+1)*a(n+1)*x^n + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 25*x^2 + 121*x^3 + 561*x^4 + 2477*x^5 + 10361*x^6 + 40817*x^7 + 150433*x^8 + 515605*x^9 + 1646041*x^10 + ...
which equals (sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2) / (2*x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..520

Crossrefs

Cf. A302701, A302704, A303064.

Programs

Maple

ogf := RootOf(16*x^2*(4*x-1)*y^3-8*x*(3*x-1)*y^2+(9*x+1)*(3*x-1)*y+1, y)^(1/2);
gfun[seriestolist](series(ogf, x=0, 31))[]; # Mark van Hoeij, Nov 28 2024

PARI

{a(n) = my(A=1); for(i=1,n, A = A = 1 + intformal( (x*A^4)'/(x*A +x*O(x^n))' );); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f. A(x) satisfies:

(1) A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.

(2) A(x) = 1 + Integral A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)) dx.

(3) A(x) = 1 + Integral A(x) * ( sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2 ) / (2*x) dx.

(4) 0 = A(x)^4 - A(x)*(1 - 4*x*A(x)^2)*A'(x) - x*A'(x)^2.