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A278079 Expansion of e.g.f. (1/3!)*sin^3(x)/cos(x) (coefficients of odd powers only).

%I A278079 #21 Feb 16 2025 08:33:37
%S A278079 0,1,0,56,1280,59136,3727360,317295616,34977546240,4848147562496,
%T A278079 825249675345920,169237314418507776,41153580031698534400,
%U A278079 11708600267324004499456,3853197364634932928839680,1452327126187528216207425536,621567950620088261848869109760
%N A278079 Expansion of e.g.f. (1/3!)*sin^3(x)/cos(x) (coefficients of odd powers only).
%H A278079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerPolynomial.html">Euler Polynomial</a>.
%F A278079 a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/3!*sin^3(x)/cos(x) ).
%F A278079 a(n) = (-1)^n*( 2/3*4^n*(4^(n+1) - 1)*Bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ).
%F A278079 a(n) = (-1)^(n+1)/(2^3*3!) * 2^(2*n+1)*( E(2*n+1,2) - 3*E(2*n+1,1) + 3*E(2*n+1,0) - E(2*n+1,-1) ), where E(n,x) is the Euler polynomial of order n.
%F A278079 a(n) = (-1)^(n+1)/8 * Sum_{k = 0..n} (9^(n-k) - 1)*binomial(2*n+1,2*k)*2^(2*k)* E(2*k, 1/2).
%F A278079 G.f. 1/3!*sin^3(x)/cos(x) = x^3/3! + 56*x^7/7! + 1280*x^9/9! + 59136*x^11/11! + ....
%p A278079 seq((-1)^n*( 2/3*4^n*(4^(n+1) - 1)*bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ), n = 0..20);
%Y A278079 Cf. A000182, A000364, A004174, A024235, A278080, A278194, A278195.
%K A278079 nonn,easy
%O A278079 0,4
%A A278079 _Peter Bala_, Nov 10 2016