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 A299433 Denominators of coefficients in S(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). 6
 1, 1, 1, 3, 6, 90, 810, 15120, 68040, 24494400, 1020600, 12933043200, 9093546000, 14122883174400, 2482538058000, 76263569141760000, 59580913392000, 15557768104919040000, 14357510604637200000, 28377369023372328960000, 8183781044643204000000, 3539793011975464314470400000, 270064774473225732000000, 13677760198273194111113625600000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS 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.: 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 + ... Related power series begin: 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 + ... 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; c[x_] = Assuming[x > 0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms]; Integrate[2*x*Sqrt[c[x]] + O[x]^terms, x] // 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(S, n))} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A299432 (numerators in S), A299430/A299431 (C), A005447/A005446 (sqrt(C). Sequence in context: A213138 A157197 A211896 * A036286 A084008 A092680 Adjacent sequences:  A299430 A299431 A299432 * A299434 A299435 A299436 KEYWORD nonn,frac AUTHOR Paul D. Hanna, Feb 09 2018 STATUS approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)