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

0, 1, 0, 56, 1280, 59136, 3727360, 317295616, 34977546240, 4848147562496, 825249675345920, 169237314418507776, 41153580031698534400, 11708600267324004499456, 3853197364634932928839680, 1452327126187528216207425536, 621567950620088261848869109760

Offset

0,4

Links

Table of n, a(n) for n=0..16.

Eric Weisstein's World of Mathematics, Euler Polynomial.

Formula

a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/3!*sin^3(x)/cos(x) ).

a(n) = (-1)^n*( 2/3*4^n*(4^(n+1) - 1)*Bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ).

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.

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).

G.f. 1/3!*sin^3(x)/cos(x) = x^3/3! + 56*x^7/7! + 1280*x^9/9! + 59136*x^11/11! + ....

Maple

seq((-1)^n*( 2/3*4^n*(4^(n+1) - 1)*bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ), n = 0..20);

Crossrefs

Cf. A000182, A000364, A004174, A024235, A278080, A278194, A278195.

Sequence in context: A219371 A183587 A219469 * A025597 A034202 A160290

Adjacent sequences: A278076 A278077 A278078 * A278080 A278081 A278082

Keyword

nonn,easy

Author

Peter Bala, Nov 10 2016

Status

approved