A299432 Numerators 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).

0, 0, 1, 2, 1, 1, -1, 1, 1, -139, 1, -571, -281, 163879, -5221, 5246819, 5459, -534703531, 91207079, -4483131259, -2650986803, 432261921612371, -6171801683, 6232523202521089, 4283933145517, -25834629665134204969, 11963983648109, -1579029138854919086429, -208697624924077, 746590869962651602203151, -29320119130515566117

Offset

0,4

Links

Table of n, a(n) for n=0..30.

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] & // Numerator (* 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) ) ); numerator(polcoeff(S, n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A299433 (denominators in S), A299430/A299431 (C), A005447/A005446 (sqrt(C)).

Sequence in context: A114171 A154243 A326698 * A252911 A176820 A343776

Adjacent sequences: A299429 A299430 A299431 * A299433 A299434 A299435

Keyword

sign,frac

Author

Paul D. Hanna, Feb 09 2018

Status

approved