A296728 Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).

1, -2, -16, 8, 12672, 571264, -44351360, -12355211520, -452681248768, 478190483394560, 132554796040912896, -18854516962334277632, -27186884683859043123200, -5502410397289951851773952, 6273206188133923322747420672, 5389680791235134726930445369344

Offset: 0

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

• sign

Examples

			arcsin(x*cos(x)) = x/1! - 2*x^3/3! - 16*x^5/5! + 8*x^7/7! + 12672*x^9/9! + ...
		

Crossrefs

Cf. A001818, A009015, A009016, A009446, A009447, A009633, A009634, A012495, A012780, A101928, A191512, A296464, A296466, A296679, A296680, A296729, A296730, A296731, A296740.

Programs

Mathematica

nmax = 16; Table[(CoefficientList[Series[ArcSin[x Cos[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

PARI

first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

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