A318599
E.g.f. A(x) satisfies: A(x) = sin(x) + cos(x)*A(x)^2 with A(0)=1.
Original entry on oeis.org
1, -1, -1, -11, -95, -1321, -22561, -474851, -11785535, -337650001, -10962505921, -397804232891, -15954963362975, -700861670953081, -33464274136282081, -1725656338796874131, -95578727098089100415, -5658893822397686566561, -356659432609686011399041, -23841281202421071709150571
Offset: 0
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S:= series((1 + sqrt(1-2*sin(2*x)))/(2*cos(x)), x, 51):
seq(coeff(S,x,j)*j!,j=0..50);
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m = 20; A[x_] = (1 + Sqrt[1 - 2 Sin[2x]] )/(2 Cos[x]); Range[0, m-1]! * CoefficientList[A[x] + O[x]^m, x] (* Jean-François Alcover, Apr 29 2019 *)
A318003
E.g.f. A(x) satisfies: A(x) = sinh(x) + cosh(x)*A(x)^2.
Original entry on oeis.org
1, 2, 13, 140, 2041, 38222, 874693, 23644280, 737301361, 26053773242, 1028890245373, 44906842244420, 2146597351615081, 111529829156824262, 6258181131400784053, 377167403797348584560, 24298520283720455935201, 1666382133585488471159282, 121205126078549481910218733, 9319638200814732292237048700
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2041*x^5/5! + 38222*x^6/6! + 874693*x^7/7! + 23644280*x^8/8! + 737301361*x^9/9! + 26053773242*x^10/10! + ...
such that A(x) = sinh(x) + cosh(x)*A(x)^2.
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{a(n) = my(A = 2*sinh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
A318004
E.g.f.: 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))).
Original entry on oeis.org
1, 1, 3, 23, 249, 3601, 65163, 1420103, 36240369, 1060638241, 35030837523, 1289122462583, 52311218246889, 2320745189970481, 111753587921091483, 5805372695984119463, 323619830261141155809, 19269740737912396000321, 1220661620760214878827043, 81966845938603736180310743, 5815923258824904181135143129
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 249*x^4/4! + 3601*x^5/5! + 65163*x^6/6! + 1420103*x^7/7! + 36240369*x^8/8! + 1060638241*x^9/9! + ...
such that
A(x) = cos(x) + sin(x)*A(x)^2.
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 16*x^3/3! + 160*x^4/4! + 2240*x^5/5! + 39392*x^6/6! + 841216*x^7/7! + 21130240*x^8/8! + 610734080*x^9/9! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 64*x^3/3! + 736*x^4/4! + 11072*x^5/5! + 206528*x^6/6! + 4607104*x^7/7! + 119766016*x^8/8! + ...
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{a(n) = my(A = 2*cos(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sin(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-3 of 3 results.