A379083 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^3)^3 ).

1, 3, 12, 58, 321, 1941, 12405, 82188, 558567, 3870694, 27245268, 194269872, 1400352702, 10187886330, 74710928103, 551676261727, 4098401671788, 30610414484517, 229717037309281, 1731295701244008, 13098454442320593, 99444838611953627, 757393732018935552, 5785220154325055826

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( Sum_{k>=1} A379087(k) * x^k/k ).

(2) A(x) = ( (1 + x*A(x)) * (1 + x^3*A(x)^(10/3)) )^3.

(3) A(x) = B(x)^3 where B(x) is the g.f. of A379090.

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

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

Prog

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

Crossrefs

Cf. A198888, A379082.

Cf. A379087, A379090.

Sequence in context: A369403 A291488 A090363 * A115086 A258173 A196708

Adjacent sequences: A379080 A379081 A379082 * A379084 A379085 A379086

Keyword

nonn

Author

Seiichi Manyama, Dec 15 2024

Status

approved