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

%I A366827 #14 Nov 25 2023 14:29:29
%S A366827 0,1,128,731,2160,4765,8896,14903,23136,33945,47680,64691,85328,
%T A366827 109941,138880,172495,211136,255153,304896,360715,422960,491981,
%U A366827 568128,651751,743200,842825,950976,1068003,1194256,1330085,1475840,1631871,1798528,1976161,2165120,2365755,2578416
%N A366827 -a(n)/7! is the coefficient of x^7 in the Taylor expansion of the k-th iteration of sin(x).
%C A366827 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)))...).
%H A366827 Jianing Song, <a href="/A366827/b366827.txt">Table of n, a(n) for n = 0..10000</a>
%H A366827 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A366827 a(n) = binomial(n,1) + 126*binomial(n,2) + 350*binomial(n,3) = (175*n^2 - 336*n + 164)*n/3. See A366834.
%F A366827 G.f.: x/(1-x)^2 + 126*x^2/(1-x)^3 + 350*x^3/(1-x)^4.
%e A366827 sin(sin(x)) = x - 2*x^3/3! + 12*x^5/5! - 128*x^7/7! + ...;
%e A366827 sin(sin(sin(x))) = x - 3*x^3/3! + 33*x^5/5! - 731*x^7/7! + ...;
%e A366827 sin(sin(sin(sin(x)))) = x - 4*x^3/3! + 64*x^5/5! - 2160*x^7/7! + ....
%o A366827 (PARI) a(n) = (175/3)*n^3 - 112*n^2 + (164/3)*n
%Y A366827 Cf. A366834 (main sequence), A051624 (coefficient of x^5), A285018, A285019.
%K A366827 nonn,easy
%O A366827 0,3
%A A366827 _Jianing Song_, Oct 25 2023