A381830 -id:A381830 - cp's OEIS frontend

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

Seiichi Manyama, Mar 07 2025

nonn

Cf. A381817, A381819, A381820.
Cf. A364592, A381830, A381831.
Cf. A000108, A381772.

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

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).

A381831 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^3 ) )^(1/3).

Original entry on oeis.org

1, 2, 14, 133, 1456, 17306, 217066, 2827896, 37895130, 519000037, 7232429952, 102220846756, 1461817707558, 21112968248198, 307527937374182, 4512344039147420, 66634574697351360, 989569163283434676, 14769533757869187052, 221426909287107012800, 3333042591222552282784, 50353576994047154278451

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A364592, A381818, A381830.
Cf. A129442, A381828.
Cf. A000108.

my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^3)/x)^(1/3))

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

a(n) = Sum_{k=0..n} binomial(3*n+k+1,k) * binomial(4*n-2*k,n-k)/(3*n+k+1).

cp's OEIS Frontend

A381818 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

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