A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

1, 2, 11, 83, 727, 6940, 70058, 735502, 7949031, 87851819, 988307647, 11279719247, 130286197186, 1520108988221, 17889102534329, 212095541328931, 2531001870925559, 30376237591559863, 366417240105654587, 4440000077166319993, 54020150448778625847, 659665548217188211288

Offset: 0

Author

Seiichi Manyama, Mar 07 2025

Keywords

• nonn

Crossrefs

Cf. A054727, A060941, A381773, A381774, A381775.

Cf. A007852, A069271, A381780.

Cf. A000108, A274052.

Programs

PARI

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

Formula

G.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * C(x*A(x)^2).

a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(2*n+1,n-k)/(2*n+2*k+1).