A365694 G.f. A(x) satisfies A(x) = 1 + x^3*A(x)^2 / (1 - x*A(x)).

1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110

Offset

0,7

Links

Robert Israel, Table of n, a(n) for n = 0..2745

Formula

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

G.f.: A(x) = 2/(1 + x + sqrt(1 + x*(-2 + x - 4*x^2))). - Vaclav Kotesovec, Sep 16 2023

D-finite with recurrence: (6 + 4*n)*a(n) + (-3 - n)*a(1 + n) + (15 + 6*n)*a(n + 2) + (-9 - 2*n)*a(n + 3) + (9 + 2*n)*a(n + 4) + (-6 - n)*a(n + 5) = 0. - Robert Israel, May 18 2026

Maple

f:= gfun:-rectoproc({(6 + 4*n)*a(n) + (-3 - n)*a(1 + n) + (15 + 6*n)*a(n + 2) + (-9 - 2*n)*a(n + 3) + (9 + 2*n)*a(n + 4) + (-6 - n)*a(n + 5), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 18 2026

Mathematica

CoefficientList[Series[2/(1 + x + Sqrt[1 + x*(-2 + x - 4*x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2023 *)

Prog

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

Crossrefs

Cf. A023426, A023432, A054514, A114997, A365695.

Cf. A365243.

Sequence in context: A178850 A018171 A306357 * A122628 A343386 A068865

Adjacent sequences: A365691 A365692 A365693 * A365695 A365696 A365697

Keyword

nonn

Author

Seiichi Manyama, Sep 16 2023

Status

approved