A317135 G.f.: Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).

1, 1, 4, 18, 92, 504, 2897, 17235, 105233, 655687, 4152461, 26650675, 172961053, 1133152365, 7484233381, 49780558057, 333155274455, 2241793462963, 15158132783457, 102938343190333, 701783155862751, 4801340686195787, 32954688422181493, 226853921031126233, 1565828056187052419, 10834714749540715871, 75142241780769151970, 522240807596491906516

Offset

0,3

Comments

Equals the self-convolution square-root of A317134.

Links

Table of n, a(n) for n=0..27.

Formula

G.f. A(x) satisfies:

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

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

(3) A(x) = Sum_{n>=0} binomial(4*n+2, n)/(2*n+1) * x^(n+1)/(1+x)^(2*n+1).

a(n) ~ 37^(1/4) * (101 + 16*sqrt(37))^(n + 1/2) / (4*sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Jul 22 2018

Example

G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 504*x^5 + 2897*x^6 + 17235*x^7 + 105233*x^8 + 655687*x^9 + 4152461*x^10 + ...
such that
A(x) = 1/(1+x) + 2*x/(1+x)^3 + 9*x^2/(1+x)^5 + 52*x^3/(1+x)^7 + 340*x^4/(1+x)^9 + 2394*x^5/(1+x)^11 + ... + A069271(n)*x^n/(1+x)^(2*n+1) + ...

Mathematica

CoefficientList[Sqrt[1/x * InverseSeries[Series[4*x/((1 + x)^2 + Sqrt[(1 + x)^4 - 4*x])^2, {x, 0, 30}], x]], x] (* Vaclav Kotesovec, Jul 22 2018 *)

Prog

(PARI) {a(n) = my(A = sum(m=0, n, binomial(4*m+2, m)/(2*m+1) * x^m / (1+x +x*O(x^n))^(2*m+1))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = sqrt( (1/x) * serreverse( 4*x/((1+x)^2 + sqrt( (1+x)^4 - 4*x  + x*O(x^n)))^2 ))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A317134.

Sequence in context: A206639 A367875 A172964 * A196150 A185298 A255397

Adjacent sequences: A317132 A317133 A317134 * A317136 A317137 A317138

Keyword

nonn

Author

Paul D. Hanna, Jul 22 2018

Status

approved