A378151 -id:A378151 - cp's OEIS frontend

A378152 G.f. A(x) satisfies A(x) = 1 + (x * (1+x) * A(x))^4.

1, 0, 0, 0, 1, 4, 6, 4, 5, 32, 112, 224, 302, 488, 1564, 4872, 11034, 19664, 37128, 95824, 266659, 635740, 1306682, 2706524, 6503711, 16794992, 40634744, 90066416, 197648134, 465436936, 1152867388, 2790870536, 6434526866, 14640368240, 34415925816, 83509570992

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A000045, A256169, A378151.

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

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

A378153 G.f. A(x) satisfies A(x) = 1 + (x * (1+x))^3 * A(x)^2.

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 3, 12, 30, 45, 75, 192, 436, 798, 1554, 3542, 7740, 15543, 32183, 70794, 153252, 321431, 684123, 1491504, 3232672, 6928779, 14957787, 32615388, 70991040, 153985890, 335256886, 733206840, 1603258134, 3503385568, 7671749664, 16837946850

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A115055, A378151.

Cf. A000108.

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

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * C(k), where C(k) are the Catalan numbers (A000108).

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

cp's OEIS Frontend

A378152 G.f. A(x) satisfies A(x) = 1 + (x * (1+x) * A(x))^4.

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