A296939 -id:A296939

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A296940

Expansion of e.g.f. sech(x*tan(x/2)) (even powers only).

+10
3

1, 0, -3, -15, 406, 14355, -189123, -42283696, -837846615, 284972761557, 28521503291230, -3070544172379761, -1054107683427761463, 1143265731049052000, 54900209444888714822181, 7959249060310612253252265, -3679623847504649619798598778, -1631286181830482909037469295781

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n) = (2*n)! * [x^(2*n)] sech(x*tan(x/2)).

EXAMPLE

sech(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! + 406*x^8/8! + ...

MATHEMATICA

nmax = 17; Table[(CoefficientList[Series[Sech[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

CROSSREFS

Cf. A000364, A001469, A009011, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Dec 22 2017

STATUS

approved

A296941

Expansion of e.g.f. arcsin(x*tan(x/2)) (even powers only).

+10
1

0, 1, 1, 18, 227, 12125, 542448, 55071205, 5492843269, 905996551626, 159770279801855, 39299019878991521, 10721872262093222016, 3707660329253983397113, 1438816154956071399594457, 668949924061617421125859650, 348908555505788456739965412203

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

a(n) = (2*n)! * [x^(2*n)] arcsin(x*tan(x/2)).

EXAMPLE

arcsin(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + ...

MATHEMATICA

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

CROSSREFS

Cf. A001469, A001818, A012780, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296942.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Dec 22 2017

STATUS

approved

A296942

Expansion of e.g.f. arcsinh(x*tan(x/2)) (even powers only).

+10
1

0, 1, 1, -12, -193, 5195, 397248, -9589391, -3147743231, -10931156748, 65632780196255, 4713930109297211, -2846093176389647904, -606335605925899344287, 213167747093485780707937, 109460864600185764327567060, -21782399212761670190907400897

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

FORMULA

a(n) = (2*n)! * [x^(2*n)] arcsinh(x*tan(x/2)).

EXAMPLE

arcsinh(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! + ...

MATHEMATICA

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

CROSSREFS

Cf. A000364, A001469, A001818, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296941.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Dec 22 2017

STATUS

approved

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