A071726 Expansion of (1+x^3*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 1, 2, 6, 16, 47, 146, 471, 1562, 5291, 18226, 63648, 224808, 801686, 2882452, 10437745, 38032110, 139339635, 512996370, 1896907980, 7041759120, 26233530210, 98046684060, 367525880670, 1381386710964, 5205006015102, 19657257219476, 74395497317456

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.

Formula

a(n) ~ 17 * 2^(2*n-4) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 20 2025

Maple

gf:= (C-> (1+x^3*C(x))*C(x))(x-> (1-sqrt(1-4*x))/(2*x)):
gfun[seriestolist](series(gf, x, 31))[];  # Alois P. Heinz, Dec 15 2025

Mathematica

With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[# (1 + x^3*#) &[(1 - (1 - 4 x)^(1/2))/(2 x)], {x, 0, 25}], x]] (* Michael De Vlieger, May 30 2020 *)

Prog

(SageMath)
x = LazyPowerSeriesRing(QQ, 'x').gen()
C = (1 - (1 - 4 * x)**(1/2)) / (2 * x)
f = (1 + x**3 * C) * C
f[:10] # F. Chapoton, Nov 26 2025

Crossrefs

Cf. A000108.

Sequence in context: A003291 A148442 A190729 * A391029 A148443 A148444

Adjacent sequences: A071723 A071724 A071725 * A071727 A071728 A071729

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved