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A299431
Denominators 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).
7
1, 1, 3, 18, 270, 432, 17010, 2721600, 40824, 1175731200, 1515591000, 217275125760, 354648294000, 5084237942784000, 114578679600, 915162829701120000, 1595278956070800000, 298709147614445568000, 818378104464320400000, 168561571998831634022400000, 982053725357184480000, 45745017385529077294694400000, 279517041579788632620000000
OFFSET
0,3
COMMENTS
Given g.f. C(x) = Sum_{n>=0} A299430(n)/A299431(n)*x^n, then C(x)^(1/2) = Sum_{n>=0} A005447(n)/A005446(n)*x^n.
FORMULA
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.
EXAMPLE
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 + ...
MATHEMATICA
terms = 30; Assuming[x>0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms] // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Feb 22 2018 *)
PROG
(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) ) ); denominator(polcoeff(C, n))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A299430 (numerators in C), A299432/A299433 (S), A005447/A005446 (sqrt(C).
Sequence in context: A232916 A327231 A279490 * A222790 A128775 A102100
KEYWORD
nonn,frac
AUTHOR
Paul D. Hanna, Feb 09 2018
STATUS
approved