A396310 a(n)/9! is the coefficient of x^9 in the Taylor expansion of the k-th iteration of sin(x).

0, 1, 1872, 25857, 121600, 368145, 873936, 1776817, 3244032, 5472225, 8687440, 13145121, 19130112, 26956657, 36968400, 49538385, 65069056, 83992257, 106769232, 133890625, 165876480, 203276241, 246668752, 296662257, 353894400, 419032225, 492772176, 575840097, 668991232

Offset

0,3

Comments

See A366834 for more information.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = binomial(n,1) + 1870*binomial(n,2) + 20244*binomial(n,3) + 29400*binomial(n,4) = 1225*n^4 - 3976*n^3 + 4288*n^2 - 1536*n.

G.f.: x/(1-x)^2 + 1870*x^2/(1-x)^3 + 20244*x^3/(1-x)^4 + 29400*x^4/(1-x)^5.

E.g.f.: exp(x)*x*(1 + 935*x + 3374*x^2 + 1225*x^3).

Example

sin(sin(x)) = x - 2*x^3/3! + 12*x^5/5! - 128*x^7/7! + 1872*x^9/9! - ...;
sin(sin(sin(x))) = x - 3*x^3/3! + 33*x^5/5! - 731*x^7/7! + 25857*x^9/9! - ...;
sin(sin(sin(sin(x)))) = x - 4*x^3/3! + 64*x^5/5! - 2160*x^7/7! + 121600*x^9/9! - ....

Mathematica

A396310[n_] := (8 - 7*n)^2*n*(25*n - 24); Array[A396310, 30, 0] (* Paolo Xausa, Jun 03 2026 *)
(* Alternative: *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 1872, 25857, 121600}, 30] (* Paolo Xausa, Jun 03 2026 *)

Prog

(PARI) a(n) = 1225*n^4 - 3976*n^3 + 4288*n^2 - 1536*n

Crossrefs

Cf. A051624 (coefficient of x^5), A366827 (coefficient of x^7).

Row 4 of A366834.

Sequence in context: A054817 A270244 A154675 * A068281 A139668 A244019

Adjacent sequences: A396307 A396308 A396309 * A396311 A396312 A396313

Keyword

nonn,easy

Author

Jianing Song, May 21 2026

Status

approved