0,3

a(n)/7! is the coefficient of x^7 in the Taylor expansion of the k-th iteration of sinh(x). This is most easily seen from the relation -i*sin(...sin(sin(sin(i*x)))...) = -i*sin(...sin(sin(i*sinh(x)))...) = -i*sin(...sin(i*sinh(sinh(x)))...) = ... = sinh(...sinh(sinh(sinh(x)))...).

Jianing Song, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

a(n) = binomial(n,1) + 126*binomial(n,2) + 350*binomial(n,3) = (175*n^2 - 336*n + 164)*n/3. See A366834.

G.f.: x/(1-x)^2 + 126*x^2/(1-x)^3 + 350*x^3/(1-x)^4.

From Elmo R. Oliveira, Sep 19 2025: (Start)

E.g.f.: exp(x)*x*(3 + 189*x + 175*x^2)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

sin(sin(x)) = x - 2*x^3/3! + 12*x^5/5! - 128*x^7/7! + ...;

sin(sin(sin(x))) = x - 3*x^3/3! + 33*x^5/5! - 731*x^7/7! + ...;

sin(sin(sin(sin(x)))) = x - 4*x^3/3! + 64*x^5/5! - 2160*x^7/7! + ....

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 128, 731}, 37] (* Amiram Eldar, Mar 28 2026 *)

(PARI) a(n) = (175/3)*n^3 - 112*n^2 + (164/3)*n

Cf. A366834 (main sequence), A051624 (coefficient of x^5), A285018, A285019.

Sequence in context: A218903 A333584 A349110 * A297463 A297700 A218070

Adjacent sequences: A366824 A366825 A366826 * A366828 A366829 A366830

nonn,easy

Jianing Song, Oct 25 2023

approved