A366827 -a(n)/7! is the coefficient of x^7 in the Taylor expansion of the k-th iteration of sin(x).

0, 1, 128, 731, 2160, 4765, 8896, 14903, 23136, 33945, 47680, 64691, 85328, 109941, 138880, 172495, 211136, 255153, 304896, 360715, 422960, 491981, 568128, 651751, 743200, 842825, 950976, 1068003, 1194256, 1330085, 1475840, 1631871, 1798528, 1976161, 2165120, 2365755, 2578416

Offset

0,3

Comments

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)))...).

Links

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

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

Formula

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.

Example

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! + ....

Prog

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

Crossrefs

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

Keyword

nonn,easy

Author

Jianing Song, Oct 25 2023

Status

approved