A296728
Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).
Original entry on oeis.org
1, -2, -16, 8, 12672, 571264, -44351360, -12355211520, -452681248768, 478190483394560, 132554796040912896, -18854516962334277632, -27186884683859043123200, -5502410397289951851773952, 6273206188133923322747420672, 5389680791235134726930445369344
Offset: 0
arcsin(x*cos(x)) = x/1! - 2*x^3/3! - 16*x^5/5! + 8*x^7/7! + 12672*x^9/9! + ...
Cf.
A001818,
A009015,
A009016,
A009446,
A009447,
A009633,
A009634,
A012495,
A012780,
A101928,
A191512,
A296464,
A296466,
A296679,
A296680,
A296729,
A296730,
A296731,
A296740.
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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}]
-
first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017
A296729
Expansion of e.g.f. arcsin(x*cosh(x)) (odd powers only).
Original entry on oeis.org
1, 4, 44, 1912, 156816, 21506816, 4420845376, 1271132964480, 487161448339712, 239980527068474368, 147742478026391141376, 111153314734461183924224, 100339775128577885016985600, 107037870347952811373977239552, 133204585741561810426003651444736
Offset: 0
arcsin(x*cosh(x)) = x/1! + 4*x^3/3! + 44*x^5/5! + 1912*x^7/7! + 156816*x^9/9! + ...
Cf.
A001818,
A009015,
A009016,
A009446,
A009447,
A009633,
A009634,
A012495,
A012780,
A101928,
A191512,
A296464,
A296466,
A296679,
A296680,
A296728,
A296730,
A296731,
A296740.
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nmax = 15; Table[(CoefficientList[Series[ArcSin[x Cosh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
-
first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cosh(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017
A013518
Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x)-cot(x)) = x/2 + x^3/16 + 3*x^5/256 + 83*x^7/30720 + 8887*x^9/12386304 + ...
Original entry on oeis.org
1, 1, 3, 83, 8887, 57539, 2419601, 298733192941, 84896691713, 54207578317691, 535009143553922969, 303988210353762448529, 39439620915967757710853, 18146112662693896499335287481
Offset: 0
Patrick Demichel (patrick.demichel(AT)hp.com)
-
Numerator[Take[CoefficientList[Series[ArcSin[Csc[x]-Cot[x]],{x,0,30}], x],{2,-1,2}]] (* Harvey P. Dale, Feb 02 2012 *)
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a(n):=(sum((binomial(2*k,k)*sum(binomial(j+2*k,2*k)*(j+2*k+1)!*2^(-4*k-j-1)*(-1)^(n+k+j)*stirling2(2*n+1,j+2*k+1),j,0,2*n-2*k))/(2*k+1),k,0,n))/(2*n+1)!; /* Vladimir Kruchinin, May 31 2013 */
A296941
Expansion of e.g.f. arcsin(x*tan(x/2)) (even powers only).
Original entry on oeis.org
0, 1, 1, 18, 227, 12125, 542448, 55071205, 5492843269, 905996551626, 159770279801855, 39299019878991521, 10721872262093222016, 3707660329253983397113, 1438816154956071399594457, 668949924061617421125859650, 348908555505788456739965412203
Offset: 0
arcsin(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + ...
Cf.
A001469,
A001818,
A012780,
A110501,
A296839,
A296841,
A296842,
A296853,
A296854,
A296856,
A296939,
A296940,
A296942.
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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}]
Showing 1-4 of 4 results.
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