A379023 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^3 ).

1, 6, 57, 653, 8277, 111780, 1576671, 22955298, 342377304, 5204438258, 80334470136, 1255798641861, 19840021268937, 316286673287724, 5081503084814883, 82193597974971157, 1337397202150986387, 21875767255039745856, 359499909751084059372, 5932767953991599086905

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Index entries for reversions of series

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( Sum_{k>=1} A379025(k) * x^k/k ).

(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(4/3)) )^3.

(3) A(x) = B(x)^3 where B(x) is the g.f. of A239107.

a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(3*(n+1)).

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

Prog

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

Crossrefs

Cf. A007863, A379021, A379024.

Cf. A239107, A379025.

Sequence in context: A130565 A124556 A369514 * A365816 A207412 A362167

Adjacent sequences: A379019 A379021 A379022 * A379024 A379025 A379026

Keyword

nonn,new

Author

Seiichi Manyama, Dec 14 2024

Status

approved