A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

1, 0, 0, 2, 2, 0, 6, 12, 6, 20, 60, 60, 90, 280, 420, 532, 1330, 2520, 3444, 6804, 14112, 21912, 37884, 77616, 133914, 223080, 432432, 793364, 1341912, 2471040, 4629196, 8076640, 14453010, 26960232, 48308832, 85794852, 157947816, 287413152, 512697900, 933072064

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

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

From Vaclav Kotesovec, Mar 23 2023: (Start)

Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 4*(n-2)*a(n-4).

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.835086681639635368143322042736678753... is the positive real root of the equation d^4 - 4*d - 4 = 0 and c = 0.2982650309662120181812121016104223... is the largest real root of the equation 1 - 20*c + 132*c^2 - 364*c^3 + 364*c^4 = 0. (End)

Mathematica

Table[Sum[Binomial[2*k, k] * Binomial[k, n-3*k], {k, 0, n/3}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 23 2023 *)

Prog

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

Crossrefs

Cf. A006139, A115962.

Cf. A360267, A361727.

Sequence in context: A323777 A292317 A285675 * A219859 A366230 A168615

Adjacent sequences: A361485 A361486 A361487 * A361489 A361490 A361491

Keyword

nonn

Author

Seiichi Manyama, Mar 22 2023

Status

approved