A372139 -id:A372139 - cp's OEIS frontend

A372137 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x)^2 ).

1, 1, 6, 21, 181, 771, 7728, 34689, 385632, 1732971, 21041598, 92147697, 1217109951, 5099210686, 73380609681, 289623783084, 4564472639880, 16722146775195, 290985244619874, 974044248064611, 18925364858562927, 56848541164586820, 1251693011560795635

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

Cf. A364475, A372139.

Cf. A372124.

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

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

A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x) ).

Original entry on oeis.org

1, 1, 4, 11, 48, 174, 784, 3219, 14816, 65082, 304656, 1393854, 6617184, 31086556, 149336672, 714494467, 3466785216, 16808037474, 82244904016, 402770823114, 1984987570016, 9797722907684, 48581811550112, 241324198117678, 1202874359046464, 6006605345531268

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

Cf. A372139.

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

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

G.f.: A(x) = 2/(1 + sqrt(1-4*x*sqrt(1+4*x))).
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(2*k,k) * binomial(k/2,n-k)/(k+1).
D-finite with recurrence n*(n-1)*(n+1)*a(n) +2*n*(n-1)*(10*n-23)*a(n-1) +12*(n-1)*(11*n^2-64*n+83)*a(n-2) +24*(4*n^3-90*n^2+352*n-369)*a(n-3) +48*(-64*n^3+528*n^2-1433*n+1290)*a(n-4) +64*(-268*n^3+3090*n^2-11882*n+15255)*a(n-5) +192*(-208*n^3+2928*n^2-13705*n+21345)*a(n-6) -1152*(4*n-25)*(4*n-19)*(2*n-11)*a(n-7)=0. - R. J. Mathar, Apr 24 2024

cp's OEIS Frontend

A372137 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x)^2 ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

cp's OEIS Frontend

A372137 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9*x)^(1/3) * A(x)^2 ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4*x)^(1/2) * A(x) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A372137 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x)^2 ).

A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x) ).