A096664
Numerators of terms in series expansion of sin(tan(x)).
Original entry on oeis.org
1, 1, -1, -55, -143, -968167, -7000033, -571772647, -843598411471, -1263845119891, -740683182137153, -474904166544135457, 2379183287545454197, 237037449673450822122569, 155015924346163216960553093, 50568039962561468889366023, 1801162678607996830733199407999, 2359789149102567189423591182268559
Offset: 0
sin(tan(x)) = x + 1/6*x^3 - 1/40*x^5 - 55/1008*x^7 - 143/3456*x^9 + ...
-
Numerator[Take[CoefficientList[Series[Sin[Tan[x]], {x,0,40}], x], {2,-1,2}]] (* G. C. Greubel, Nov 17 2016 *)
A096671
Denominators of terms in series expansion of sin(tan(x)), odd powers only.
Original entry on oeis.org
1, 6, 40, 1008, 3456, 39916800, 566092800, 100590336000, 355687428096000, 1431118828339200, 2688996956405760000, 8617338912961658880000, 224800145555521536000000, 10888869450418352160768000000, 8841761993739701954543616000000, 4500732706172918893117440000000, 280107019961673757919941754880000000
Offset: 0
x + (1/6)*x^3 - (1/40)*x^5 - (55/1008)*x^7 - (143/3456)*x^9 + ...
A045688
Numerators of expansion of tan(sin(x)) - sin(tan(x)).
Original entry on oeis.org
0, 0, 0, 1, 29, 1913, 95, 311148869, 10193207, 1664108363, 2097555460001, 374694625074883, -48074332710505411, -4945837680716249285383, -2229960300928783674917, -24840355960694307625698893, -4153678459092607393588136503, -61444568105054860319920405585007, -335560999071136641584079760493377
Offset: 0
1/30*x^7 + 29/756*x^9 + 1913/75600*x^11 + O(x^12).
- V. I. Arnold, Huygens and Barrow, Newton and Hooke, Birkhäuser, Basel, 1990.
-
Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] (* Shows plot, which, as Bill Gosper and Neil Bickford remark, is ultraflat near 0 and has wild oscillations near Pi/2. - N. J. A. Sloane, Apr 11 2013 *)
A096718
Denominators of terms in series expansion of arcsin(arctan(x)).
Original entry on oeis.org
1, 6, 120, 5040, 362880, 4435200, 32947200, 145297152000, 69701632000, 13516122267648000, 5676771352412160000, 2872446304320552960000, 1723467782592331776000000, 1935799013407707050803200, 485144691014524112732160000, 1353553687930522274522726400000, 204193242064947360270857011200000
Offset: 0
arcsin(arctan(x)) = x - 1/6*x^3 + 13/120*x^5 - 341/5040*x^7 + 18649/362880*x^9 - 177761/4435200*x^11 + ...
-
Denominator[Take[CoefficientList[Series[ArcSin[ArcTan[x]], {x,0,40}], x], {2,-1,2}]] (* G. C. Greubel, Nov 17 2016 *)
A096717
Numerators of terms in series expansion of arcsin(arctan(x)).
Original entry on oeis.org
1, -1, 13, -341, 18649, -177761, 1087433, -4043494549, 1674761567, -284891766539657, 106410874319325461, -48402125366670946877, 26344930021064765797249, -27048608191991004321089, 6237195766537863970288933, -16102066950215127630856787159, 2258820895862623437612519923989
Offset: 0
arcsin(arctan(x)) = x - 1/6*x^3 + 13/120*x^5 - 341/5040*x^7 + 18649/362880*x^9 - 177761/4435200*x^11 + ...
Cf.
A096718,
A096664,
A096671,
A096712,
A096716,
A045688,
A045689,
A096719,
A096720,
A096721,
A096722.
-
Numerator[Take[CoefficientList[Series[ArcSin[ArcTan[x]], {x,0,50}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 17 2016 *)
A096719
Numerators of terms in series expansion of arctan(arcsin(x)).
Original entry on oeis.org
1, -1, 13, -173, 12409, -123379, 29518679, -889424791, 92273231203, 3836321172631, 22487012578592981, 2865860401219263691, 35970731592390474409, 277817773865257308429491, 1687365015862907602230599, 22415401434548677685890690591, 5789220720660809183499012532793, 2838956049184596030388390046497291
Offset: 0
arctan(arcsin(x)) = x - 1/6*x^3 + 13/120*x^5 - 173/5040*x^7 + 12409/362880*x^9 - 123379/13305600*x^11 + ...
-
Numerator[Take[CoefficientList[Series[ArcTan[ArcSin[x]], {x,0,40}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 18 2016 *)
-
a(n):=b(2*n+1);
b(n):=num(1/n*sum((1-(-1)^(m))*(-1)^((m-1)/2)*(1+(-1)^(n-m))/4*sum((sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2)),j,1,k))*binomial(k+n-1,n-1),k,1,n-m),m,1,n-1)+(1-(-1)^(n))/(2)*(-1)^((n-1)/2)/n); /* Vladimir Kruchinin, May 02 2011 */
A096720
Denominators of terms in series expansion of arctan(arcsin(x)).
Original entry on oeis.org
1, 6, 120, 5040, 362880, 13305600, 2075673600, 435891456000, 13173608448000, 13516122267648000, 5676771352412160000, 2872446304320552960000, 14243535393325056000000, 241974876675963381350400000, 949196134593634133606400000, 20303305318957834117840896000000, 4288058083363894565687997235200000
Offset: 0
arctan(arcsin(x)) = x - 1/6*x^3 + 13/120*x^5 - 173/5040*x^7 + 12409/362880*x^9 - 123379/13305600*x^11 + ...
-
Denominator[Take[CoefficientList[Series[ArcTan[ArcSin[x]],{x,0,40}],x] ,{2,-1,2}]] (* Harvey P. Dale, May 04 2013 *)
A096722
Denominators of terms in series expansion of arcsin(arctan(x)) - arctan(arcsin(x)).
Original entry on oeis.org
1, 1, 1, 30, 756, 75600, 199584, 54486432000, 2421619200, 151227648000, 5913303492096000, 5203707073044480000, 512936840057241600000, 5041143264082570444800000, 1238175538546596249600000, 11695452372671563431936000000, 33500453776280426294437478400000, 44295044437526341433756221440000000
Offset: 0
arcsin(arctan(x)) - arctan(arcsin(x)) = -1/30*x^7 + 13/756*x^9 - 2329/75600*x^11 + 3749/199584*x^13 - 1405132357/54486432000*x^15 + ...
Cf.
A096721,
A096717,
A096718,
A096664,
A096671,
A096712,
A096716,
A045688,
A045689,
A096719,
A096720.
-
Denominator[Take[CoefficientList[Series[ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]], {x,0,40}], x], {2,-1,2}]] (* G. C. Greubel, Nov 18 2016 *)
A096721
Numerators of terms in series expansion of arcsin(arctan(x)) - arctan(arcsin(x)).
Original entry on oeis.org
0, 0, 0, -1, 13, -2329, 3749, -1405132357, 41223659, -3230487913, 87420689313263, -92876785811395309, 6545378422138547141, -76226954122169434345117, 13717355610784766550119, -152042860419225571514252591, 325359516347299085987218014617, -501994552683503696983628163720749, 226141284010354023120430917899293
Offset: 0
arcsin(arctan(x)) - arctan(arcsin(x)) = -1/30*x^7 + 13/756*x^9 - 2329/75600*x^11 + 3749/199584*x^13 - 1405132357/54486432000*x^15 + ...
Cf.
A096722,
A096717,
A096718,
A096664,
A096671,
A096712,
A096716,
A045688,
A045689,
A096719,
A096720.
-
With[{nn=40},Numerator[Take[CoefficientList[Series[ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]],{x,0,nn}],x],{2,-1,2}]]] (* Harvey P. Dale, Dec 07 2011 *)
A096725
Numerators of terms in series expansion of (sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))).
Original entry on oeis.org
1, 5, 1313, -2773, -701933647, -86849082293, -174426488476171, -130176915706274917, -42426469007472079018663, -24495552034235134641205327, -3019410235003955483667737236843, -74265172933666226350348992663473, -2457268368880426576340457161112391, -589361165665450343618737576026916723726003
Offset: 0
(sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))) = 1 + 5/3*x^2 + 1313/1890*x^4 - 2773/11907*x^6 - 701933647/1650310200*x^8 - 86849082293/270320810760*x^10 - ...
- V. I. Arnold, Huygens and Barrow, Newton and Hooke, Birkhäuser, Basel, 1990.
Cf.
A096730,
A096722,
A096717,
A096718,
A096664,
A096671,
A096712,
A096716,
A045688,
A045689,
A096719,
A096720.
-
Numerator[Take[CoefficientList[Series[(Sin[Tan[x]] - Tan[Sin[x]]) / (ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]]), {x,0,50}], x], {1, -1, 2}]] (* G. C. Greubel, Nov 20 2016 *)
Showing 1-10 of 11 results.