A374563 Expansion of g.f. A(x) satisfying A(x) = x*(1 + A(x)^2) + x^2*(1 + A(x)^2)^2.

1, 1, 1, 4, 7, 17, 47, 112, 302, 819, 2187, 6072, 16863, 47099, 133289, 378352, 1080522, 3104302, 8950670, 25920104, 75342011, 219680831, 642547985, 1884571240, 5541269802, 16331880595, 48239191795, 142769840280, 423339407025, 1257470646765, 3741247990455, 11148083590080

Offset

1,4

Links

Paul D. Hanna, Table of n, a(n) for n = 1..600

Formula

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

(1) A(x) = x*(1 + A(x)^2) + x^2*(1 + A(x)^2)^2.

(2) A(x) = Series_Reversion( (sqrt(1 + 4*x) - 1)/(2*(1 + x^2)) ).

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

(4) C(A(x)) = x + x*A(x)^2, where C(x) = x - C(x)^2 is a g.f. of the Catalan numbers (A000108).

a(n) ~ sqrt((1 + s^2)/(2*r*s*(1 + r*(2 + 6*s^2)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.3201411821955004503644495595974372984436524828585... and s = 0.7723300090737596252395061122641790356344153664573... are positive real roots of the system of equations r*(1 + s^2)*(1 + r + r*s^2) = s, 2*r*s*(1 + 2*r + 2*r*s^2) = 1. - Vaclav Kotesovec, Jul 19 2024

Example

G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 7*x^5 + 17*x^6 + 47*x^7 + 112*x^8 + 302*x^9 + 819*x^10 + 2187*x^11 + 6072*x^12 + ...
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 23*x^6 + 56*x^7 + 158*x^8 + 408*x^9 + 1107*x^10 + 3080*x^11 + 8459*x^12 + ...
Let B(x) be the series reversion of A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 4*x^4 + 13*x^5 - 38*x^6 + 119*x^7 - 391*x^8 + ...
and B(x) = (sqrt(1 + 4*x) - 1)/(2*(1 + x^2)).

Prog

(PARI) {a(n) = my(A = serreverse( (sqrt(1 + 4*x +x^2*O(x^n)) - 1)/(2*(1 + x^2)) )); polcoeff(A, n)}
for(n=1, 32, print1(a(n), ", "))

Crossrefs

Cf. A000108.

Sequence in context: A049944 A098091 A319782 * A057450 A077274 A292850

Adjacent sequences: A374560 A374561 A374562 * A374564 A374565 A374566

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2024

Status

approved