A383569 Expansion of 1/sqrt((1-x^7)^2 - 4*x^2).

1, 0, 2, 0, 6, 0, 20, 1, 70, 6, 252, 30, 924, 140, 3433, 630, 12882, 2772, 48710, 12012, 185316, 51481, 708582, 218810, 2720788, 923990, 10484684, 3881556, 40528441, 16236486, 157086660, 67675972, 610318610, 281236620, 2376289056, 1165715161, 9269869182

Offset

0,3

Comments

Number of lattice paths from (0,0) to (n,n) using steps (2,0),(0,2),(7,7).

Diagonal of the rational function 1 / (1 - x^2 - y^2 - x^7*y^7).

Diagonal of the rational function 1 / ((1-x^2*y)*(1-x^5*y^6) - y).

Links

Robert Israel, Table of n, a(n) for n = 0..3277

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Formula

(n + 7)*a(n) + (-21 - 2*n)*a(n + 7) + (-4*n - 52)*a(n + 12) + (n + 14)*a(n + 14) = 0. - Robert Israel, Sep 09 2025

Maple

with(gfun):
g:=  1/sqrt((1-x^7)^2 - 4*x^2):
rec:= diffeqtorec(holexprtodiffeq(g, y(x)), y(x), a(n)):
f:= rectoproc(rec, a(n), remember):
map(f, [$0..50]); # Robert Israel, Sep 09 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-x^7)^2-4x^2], {x, 0, 40}], x] (* Harvey P. Dale, Aug 09 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x^7)^2-4*x^2))

Crossrefs

Cf. A002426, A053442, A383568.

Sequence in context: A369278 A126869 A094233 * A094659 A321907 A361522

Adjacent sequences: A383566 A383567 A383568 * A383570 A383571 A383572

Keyword

nonn

Author

Seiichi Manyama, Apr 30 2025

Status

approved