A372038 -id:A372038 - cp's OEIS frontend

A372087 G.f. A(x) satisfies A(x) = 1/( 1 - 9x(1 + x)*A(x) )^(1/3).

1, 3, 30, 369, 5130, 76626, 1200816, 19475829, 324140886, 5504511654, 94998663000, 1661370690546, 29377608173460, 524366947411668, 9435112261205328, 170958245619049173, 3116653690408787070, 57125853834377116014, 1052116816793294021688

Offset: 0

Seiichi Manyama, Apr 17 2024

nonn

Cf. A180400, A372038.

a(n) = sum(k=0, n, 9^k*binomial(4*k/3-2/3, k)*binomial(k, n-k)/(k+1));

a(n) = Sum_{k=0..n} 9^k * binomial(4*k/3-2/3,k) * binomial(k,n-k)/(k+1).

A372039 Expansion of ( 1 + 9x(1 + x) )^(1/3).

Original entry on oeis.org

1, 3, -6, 27, -144, 837, -5139, 32778, -215001, 1440747, -9818820, 67834665, -473945580, 3342743235, -23766448545, 170148578130, -1225477405485, 8873126329095, -64547392633740, 471509782020405, -3457212506428230, 25434642838306185, -187694935991201745

Offset: 0

Seiichi Manyama, Apr 16 2024

sign

Cf. A180400, A361843, A372024, A372038.

my(N=30, x='x+O('x^N)); Vec((1+9*x*(1+x))^(1/3))

a(n) = sum(k=0, n, 9^k*binomial(1/3, k)*binomial(k, n-k));

a(n) = Sum_{k=0..n} 9^k * binomial(1/3,k) * binomial(k,n-k).

a(n) ~ (-1)^(n+1) * Gamma(1/3) * 5^(1/6) * 3^(n - 1/2) * phi^(2*n - 2/3) / (2*Pi*n^(4/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 19 2024

cp's OEIS Frontend

A372087 G.f. A(x) satisfies A(x) = 1/( 1 - 9x(1 + x)*A(x) )^(1/3).

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A372039 Expansion of ( 1 + 9x(1 + x) )^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

cp's OEIS Frontend

A372087 G.f. A(x) satisfies A(x) = 1/( 1 - 9*x*(1 + x)*A(x) )^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A372039 Expansion of ( 1 + 9*x*(1 + x) )^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A372087 G.f. A(x) satisfies A(x) = 1/( 1 - 9x(1 + x)*A(x) )^(1/3).

A372039 Expansion of ( 1 + 9x(1 + x) )^(1/3).