A229043 Series reversion of (sqrt(1+4*x) - 1)/2 - x^3.

1, 1, 1, 5, 12, 35, 122, 390, 1320, 4631, 16185, 57707, 208348, 756840, 2775012, 10246206, 38043339, 142045387, 532888840, 2007554241, 7592537590, 28814794105, 109702322730, 418868083725, 1603584623355, 6154156653687, 23671591739306, 91242219125712, 352378515196920, 1363360128627380

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..1647

Robert Israel, Recurrence

Formula

G.f. A(x) satisfies:

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

(2) A(x)^3 = A(x)*C(-A(x)) - x^3, where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

(3) A(x) = G(x)*(1 + G(x)) where G(x) = x + A(x)^3.

a(n) ~ r^(1/2-n) / (2 * sqrt(3*Pi*s*(1+9*s^5)) * n^(3/2)), where s = 0.44683030245197... is the root of the equation 9*s^4*(1+4*s) = 1 and r = -1/2 + 1/(6*s^2) - s^3 = 0.24555068038... - Vaclav Kotesovec, Jan 22 2014

D-finite with recurrence: Sequence satisfies a 9th-order linear recurrence with coefficients that are polynomials in n of degree 5: see link. - Robert Israel, May 14 2018

Example

G.f.: A(x) = x + x^2 + x^3 + 5*x^4 + 12*x^5 + 35*x^6 + 122*x^7 + 390*x^8 +...
where the series reversion of g.f. A(x) begins:
(sqrt(1+4*x) - 1)/2 - x^3 = x - x^2 + x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +...+ (-1)^(n-1)*A000108(n-1)*x^n +...
The cube of the g.f. equals the series:
A(x)^3 = x^3*(1+x)^3 + d/dx x^6*(1+x)^6/2! + d^2/dx^2 x^9*(1+x)^9/3! + d^3/dx^3 x^12*(1+x)^12/4! + d^4/dx^4 x^15*(1+x)^15/5! +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 22*x^6 + 72*x^7 + 225*x^8 + 790*x^9 + 2739*x^10 +...
RELATED EXPANSIONS.
G.f. A(x) = G(x)*(1 + G(x)) = (G(x) - x)^(1/3) where G(x) begins:
G(x) = x + x^3 + 3*x^4 + 6*x^5 + 22*x^6 + 72*x^7 + 225*x^8 + 790*x^9 +...

Maple

with(gfun):
S:= solve((sqrt(1+4*x)-1)/2-x^3=y, x):
DE:= holexprtodiffeq(S, g(y)):
Rec:= diffeqtorec(DE, g(y), a(n)):
f:= rectoproc(Rec, a(n), remember):
map(f, [$1..40]); # Robert Israel, May 14 2018

Mathematica

Rest[CoefficientList[InverseSeries[Series[(Sqrt[1+4*x]-1)/2-x^3, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(PARI) {a(n)=polcoeff( serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^3 ), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* G.f. A(x) = sqrt(G(x) - x) where G(x) = x + G(x)^3*(1 + G(x))^3 */
{a(n)=local(G=serreverse(x-x^3*(1+x)^3+x^3*O(x^n))); polcoeff((G-x)^(1/3), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ n-th derivative
{a(n)=local(A3=x); A3=sum(m=1, n, Dx(m-1, x^(3*m)*(1+x+x*O(x^n))^(3*m)/m!)); polcoeff(A3^(1/3), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) x='x+O('x^99); Vec(serreverse(((1+4*x)^(1/2)-1)/2-x^3)) \\ Altug Alkan, May 14 2018

Crossrefs

Cf. A229042.

Sequence in context: A116995 A032281 A294654 * A185699 A301873 A368094

Adjacent sequences: A229040 A229041 A229042 * A229044 A229045 A229046

Keyword

nonn

Author

Paul D. Hanna, Nov 02 2013

Extensions

Offset corrected by Vaclav Kotesovec, Jan 22 2014

Status

approved