A263246
Expansion of e.g.f.: sin(r*x) / sqrt(1 + cos(r*x)^2) where r = sqrt(2), odd powers only.
Original entry on oeis.org
1, 1, -11, -491, -11159, 460681, 103577629, 8160790429, -624333860399, -386787409545839, -68810049201689531, 6999828208693648549, 9872674440874152431161, 3255253386897615662908441, -346248578699462435167833491, -1072454627614122049417452882131, -584579592415141205182370782224479, 47874474639430619859527348515679521
Offset: 1
E.g.f.: S(x) = x + x^3/3! - 11*x^5/5! - 491*x^7/7! - 11159*x^9/9! + 460681*x^11/11! + 103577629*x^13/13! + 8160790429*x^15/15! +...
Related expansions.
S(x)^2 = 2*x^2/2! + 8*x^4/4! - 112*x^6/6! - 9088*x^8/8! - 310528*x^10/10! + 14701568*x^12/12! +...+ -A263249(n)*x^(2*n)/(2*n)! +...
sqrt(1 - S(x)^2) = 1 - x^2/2! - 7*x^4/4! - 49*x^6/6! + 1457*x^8/8! + 148799*x^10/10! + 6409193*x^12/12! +...+ A263247(n)*x^(2*n)/(2*n)! +...
sqrt(1 + S(x)^2) = 1 + x^2/2! + x^4/4! - 71*x^6/6! - 2591*x^8/8! - 23759*x^10/10! + 7872481*x^12/12! +...+ A263248(2*n)*x^(2*n)/(2*n)! +...
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r:= Sqrt[2]; With[{nmax = 500}, CoefficientList[Series[Sin[r*x]/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax - 1]!][[2 ;; -1 ;; 2]] (* G. C. Greubel, Jul 27 2018 *)
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{a(n) = local(S=x,C=1,D=1,ox=O(x^(2*n+2))); for(i=1,2*n+1, S = intformal(C*D^2 +ox); C = 1 - intformal(S*D^2); D = 1 + intformal(S*C*D);); (2*n+1)!*polcoeff(S, 2*n+1)}
for(n=0,20,print1(a(n),", "))
A263249
Expansion of e.g.f.: 2*cos(r*x)^2 / (1 + cos(r*x)^2) where r = sqrt(2), even terms only.
Original entry on oeis.org
1, -2, -8, 112, 9088, 310528, -14701568, -4554426368, -458243735552, 37024075153408, 29290212127670272, 6224109737622372352, -631398107821314670592, -1112417825593218314534912, -422420220419591934719295488, 41942640830461258871206838272, 165285368668709582104936440659968, 101410495525765825487306697440493568
Offset: 0
E.g.f.: A(x) = 1 - 2*x^2/2! - 8*x^4/4! + 112*x^6/6! + 9088*x^8/8! + 310528*x^10/10! - 14701568*x^12/12! - 4554426368*x^14/14! +...
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With[{nn=40},Take[CoefficientList[Series[(2 Cos[x*Sqrt[2]]^2)/(1+Cos[ x*Sqrt[2]]^2),{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Mar 13 2018 *)
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{a(n) = local(S=x,C=1,D=1,ox=O(x^(2*n+2))); for(i=1,2*n+1, S = intformal(C*D^2 +ox); C = 1 - intformal(S*D^2); D = 1 + intformal(S*C*D);); (2*n)!*polcoeff(C^2, 2*n)}
for(n=0,20,print1(a(n),", "))
A263247
Expansion of e.g.f.: r*cos(r*x) / sqrt(1 + cos(r*x)^2) where r = sqrt(2), even terms only.
Original entry on oeis.org
1, -1, -7, -49, 1457, 148799, 6409193, -436948849, -155175606943, -18245982604801, 1864031151256793, 1627915037217907151, 390178889220670506257, -46784571591411151243201, -89306450512551172914577207, -37461031331532428265812712049, 4204976347690709918899169381057, 17814701962096793952255775890393599
Offset: 0
E.g.f.: C(x) = 1 - x^2/2! - 7*x^4/4! - 49*x^6/6! + 1457*x^8/8! + 148799*x^10/10! + 6409193*x^12/12! - 436948849*x^14/14! +...
Related expansions.
C(x)^2 = 1 - 2*x^2/2! - 8*x^4/4! + 112*x^6/6! + 9088*x^8/8! + 310528*x^10/10! - 14701568*x^12/12! +...+ A263249(n)*x^(2*n)/(2*n)! +...
sqrt(1 - C(x)^2) = x + x^3/3! - 11*x^5/5! - 491*x^7/7! - 11159*x^9/9! + 460681*x^11/11! +...+ A263246(n)*x^(2*n+1)/(2*n+1)! +...
sqrt(2 - C(x)^2) = 1 + x^2/2! + x^4/4! - 71*x^6/6! - 2591*x^8/8! - 23759*x^10/10! + 7872481*x^12/12! +...+ A263248(n)*x^(2*n)/(2*n)! +...
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r:= Sqrt[2]; With[{nmax = 60}, CoefficientList[Series[r*Cos[r*x]/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 27 2018 *)
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{a(n) = local(S=x,C=1,D=1,ox=O(x^(2*n+2))); for(i=1,2*n+1, S = intformal(C*D^2 +ox); C = 1 - intformal(S*D^2); D = 1 + intformal(S*C*D);); (2*n)!*polcoeff(C, 2*n)}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.