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A299430
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Numerators of coefficients in C(x) where: C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 2*x*C(x).
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7
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1, 2, 5, 13, 43, 5, -19, 41, 1, -7243, 923, 23183, -21401, 64243259, -73, -471741703, 328578883, -11162934047, -103170103, 405632121933911, -811862551, 6065578925646109, 2180051549129, -261709011005693843, 36779766732769, -1552304807519349743393, -33230318647573, 727642520296392573166003, -85602927913097260249, 3885938896026347271301517
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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The functions C = C(x) and S = S(x) satisfy:
(1) sqrt(C) - sqrt(S) = 1.
(2a) C'*sqrt(S) = S'*sqrt(C) = 2*x*C.
(2b) C' = 2*x*C/sqrt(S).
(2c) S' = 2*x*sqrt(C).
(3a) C = 1 + Integral 2*x*C/sqrt(S) dx.
(3b) S = Integral 2*x*sqrt(C) dx.
(4a) sqrt(C) = exp( Integral x/(sqrt(C) - 1) dx ).
(4b) sqrt(S) = exp( Integral x/sqrt(S) dx ) - 1.
(5a) C - S = exp( Integral 2*x*C/(C*sqrt(S) + S*sqrt(C)) dx ).
(5b) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
(6a) sqrt(C) = exp( sqrt(C) - 1 - x^2/2 ).
(6b) sqrt(C) = 1 + x^2/2 + Integral x/(sqrt(C) - 1) dx.
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EXAMPLE
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G.f.: C(x) = 1 + 2*x + 5/3*x^2 + 13/18*x^3 + 43/270*x^4 + 5/432*x^5 - 19/17010*x^6 + 41/2721600*x^7 + 1/40824*x^8 - 7243/1175731200*x^9 + 923/1515591000*x^10 + ...
Related power series begin:
S(x) = x^2 + 2/3*x^3 + 1/6*x^4 + 1/90*x^5 - 1/810*x^6 + 1/15120*x^7 + 1/68040*x^8 - 139/24494400*x^9 + 1/1020600*x^10 - 571/12933043200*x^11 + ...
sqrt(C) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 - 571/2351462400*x^9 - 281/1515591000*x^10 + ... + A005447(n)/A005446(n)*x^n + ...
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MATHEMATICA
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terms = 30; Assuming[x>0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms] // CoefficientList[#, x]& // Numerator (* Jean-François Alcover, Feb 22 2018 *)
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PROG
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(PARI) {a(n) = my(C=1, S=x^2); for(i=0, n, C = (1 + sqrt(S +O(x^(n+2))))^2; S = intformal( 2*x*sqrt(C) ) ); numerator(polcoeff(C, n))}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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