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
Programs
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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 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))
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