A381818 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, 12, 97, 903, 9129, 97419, 1080058, 12319200, 143630575, 1704099034, 20507897766, 249734145622, 3071587654688, 38102046141882, 476138815310364, 5988435287060671, 75745116484532586, 962898676577135634, 12295850972794555196, 157649023155654522723, 2028662477759375282902

Offset: 0

Author

Seiichi Manyama, Mar 07 2025

Keywords

• nonn

Crossrefs

Cf. A381817, A381819, A381820.

Cf. A364592, A381830, A381831.

Cf. A000108, A381772.

Programs

PARI

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

Formula

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

a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(3*n-k,n-k)/(2*n+2*k+1).