A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

1, 0, 2, 4, 16, 52, 188, 672, 2458, 9052, 33648, 125864, 473500, 1789632, 6791528, 25863568, 98796096, 378411332, 1452886052, 5590262688, 21551271916, 83228809640, 321933018272, 1247062996304, 4837152438556, 18785529571200, 73037938668632, 284268423472432

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

Sum_{k=0..n} a(k) * a(n-k) = A387085(n).

G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).

G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.

G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.

a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - Vaclav Kotesovec, Aug 20 2025

D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

Mathematica

nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)
CoefficientList[Series[1/Sqrt[4*x-1+2*Sqrt[1-4*x]], {x, 0, 25}], x] (* Vincenzo Librandi, Jan 20 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!  1/Sqrt( 4*x - 1 + 2*Sqrt(1 - 4*x) )); // Vincenzo Librandi, Jan 20 2026

Crossrefs

Cf. A183161, A208977, A387084.

Cf. A000108, A000984, A387085.

Sequence in context: A153948 A284730 A255730 * A363441 A087972 A010362

Adjacent sequences: A387083 A387084 A387085 * A387087 A387088 A387089

Keyword

nonn

Author

Seiichi Manyama, Aug 16 2025

Status

approved