A278194 E.g.f. (1/5!)*sin^5(x)/cos(x) (coefficients of odd powers only).

0, 0, 1, -14, 336, -1408, 256256, 14746368, 1766772736, 242121048064, 41267065061376, 8461792420167680, 2057680174397259776, 585429994601202057216, 192659868531986620481536, 72616356304572571212316672, 31078397531081274526066016256

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

Formula

a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/5!*sin^5(x)/cos(x) ).

a(n) = (-1)^n*( 4^(n-2)*(4^n - 3) + 4^(n-1)*(4^(n+1) - 1)*Bernoulli(2*n + 2)/(n + 1) )/15.

a(n) = (-1)^n/(3!*2^6) * Sum_{k = 0..n} ( 25^(n-k) - 3*9^(n-k) + 2 )*binomial(2*n+1, 2*k)*2^(2*k)*E(2*k, 1/2), where E(n,x) is the Euler polynomial of order n.

a(n) = (-1)^n/(2^5*5!) * 2^(2*n+1)*( E(2*n+1, 3) - 5*E(2*n+1, 2) + 10*E(2*n+1, 1) - 10*E(2*n+1, 0) + 5*E(2*n+1, -1) - E(2*n+1, -2) ).

G.f. 1/5!*sin^5(x)/cos(x) = x^5/5! - 14*x^7/7! + 336*x^9/9! - 1408*x^11/11! + ....

Maple

seq((-1)^n*( 4^(n-2)*(4^n - 3) + 4^(n-1)*(4^(n+1) - 1)*bernoulli(2*n + 2)/(n + 1) )/15, n = 0..20);

Prog

(PARI) a(n)={my(m=2*n+1, A=O(x*x^m)); m!*polcoef(sin(x + A)^5/cos(x + A), m)/120} \\ Andrew Howroyd, May 04 2020

Crossrefs

Cf. A000364, A004174, A024235, A278079, A278080, A278195.

Sequence in context: A277769 A060075 A222984 * A035273 A239783 A020143

Adjacent sequences: A278191 A278192 A278193 * A278195 A278196 A278197

Keyword

sign,easy

Author

Peter Bala, Nov 15 2016

Extensions

Terms a(15) and beyond from Andrew Howroyd, May 04 2020

Status

approved