A229042 Series reversion of (sqrt(1+4*x) - 1)/2 - x^2.

1, 2, 6, 25, 114, 560, 2880, 15321, 83600, 465322, 2631668, 15079922, 87362184, 510837760, 3010987912, 17870854206, 106713713826, 640659094566, 3864643224900, 23412690485800, 142386465217920, 868967571732540, 5320093500153120, 32666064906596550, 201109232686971492

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f. A(x) satisfies:

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

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

(3) A(x) = G(x)*(1 + G(x)) = sqrt(G(x) - x) where G(x) is the g.f. of A214372.

a(n) ~ r^(1/2-n) / (2*sqrt(Pi) * sqrt(1+8*s^3) * n^(3/2)), where s = ((3392 - 384*sqrt(78))^(1/3) + 4*(53 + 6*sqrt(78))^(1/3) - 4)/48 = 0.328649053... is the root of the equation 4*s^2*(1+4*s) = 1 and r = -1/2 + 1/(4*s) - s^2 = 0.152679653319... - Vaclav Kotesovec, Jan 22 2014

a(n) = binomial(2*(n-1),(n-1))/n-sum(k=1..n-1, k*(-1)^(k-n)*binomial(n+k-1,n-1)*((sum(l=0..k-1, (binomial(k-1,l)*binomial(2*(n+l-k),n-k-1))/(n+l-k)))))/n. - Vladimir Kruchinin, Feb 08 2015

D-finite with recurrence 3503*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(3377*n-3903)*a(n-1) -20*(n-2)*(2668*n^2-11692*n+12387)*a(n-2) +8*(-6616*n^3+57120*n^2-162464*n+152595)*a(n-3) -480*(4*n-15)*(4*n-13)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...
where the series reversion of g.f. A(x) begins:
(sqrt(1+4*x) - 1)/2 - x^2 = x - 2*x^2 + 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 square of the g.f. equals the series:
A(x)^2 = x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! + d^4/dx^4 x^10*(1+x)^10/5! +...
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...
Related Expansions:
G.f. A(x) = G(x) + G(x)^2 = sqrt(G(x) - x) where G(x) is the g.f. of A214372:
G(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[(Sqrt[1+4*x]-1)/2-x^2, {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^2 ), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* G.f. A(x) = sqrt(G(x) - x) where G(x) = x + G(x)^2*(1 + G(x))^2 */
{a(n)=local(G=serreverse(x-x^2*(1+x)^2+x^2*O(x^n))); polcoeff(sqrt(G-x), 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(A2=x); A2=sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(sqrt(A2), n)}
for(n=1, 30, print1(a(n), ", "))
(Maxima)
a(n):=binomial(2*(n-1), (n-1))/n-sum(k*(-1)^(k-n)*binomial(n+k-1, n-1)*((sum((binomial(k-1, l)*binomial(2*(n+l-k), n-k-1))/(n+l-k), l, 0, k-1))), k, 1, n-1)/n; /* Vladimir Kruchinin, Feb 08 2015 */

Crossrefs

Cf. A214372, A229043.

Sequence in context: A276277 A355862 A299098 * A269484 A014277 A006965

Adjacent sequences: A229039 A229040 A229041 * A229043 A229044 A229045

Keyword

nonn

Author

Paul D. Hanna, Oct 24 2013

Status

approved