A372137 -id:A372137 - cp's OEIS frontend

A372139 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x)^2 ).

1, 1, 5, 22, 131, 775, 5018, 33066, 225915, 1569663, 11114665, 79715150, 578603008, 4239698240, 31329174328, 233166912152, 1746351112863, 13152340002195, 99545267265903, 756752801436930, 5775810585852675, 44241685802126865, 339991810347999194

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

Cf. A364475, A372137.

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

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

A372124 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x) ).

Original entry on oeis.org

1, 1, 5, 8, 86, 12, 2418, -6015, 97271, -490693, 4991069, -33481184, 294850612, -2232642956, 18815166552, -150373925928, 1255171140378, -10300278908424, 86135158514634, -717384480699522, 6029697856319760, -50699911500290454, 428507430151063548

Offset: 0

Seiichi Manyama, Apr 20 2024

sign

Cf. A000108, A372137.

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+9*x)^(1/3))))

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

G.f.: A(x) = 2/(1 + sqrt(1-4*x*(1+9*x)^(1/3))).

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

cp's OEIS Frontend

A372139 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x)^2 ).

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A372124 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x) ).

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cp's OEIS Frontend

A372139 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4*x)^(1/2) * A(x)^2 ).

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A372124 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9*x)^(1/3) * A(x) ).

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PARI

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A372139 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4x)^(1/2) A(x)^2 ).

A372124 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 9x)^(1/3) A(x) ).