OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) = C(x) + S(x), where series bisections C(x) and S(x) satisfy the following relations.
(1a) S(x) = Integral C(x) * C(S(x)) dx.
(1b) C(x) = 1 + Integral S(x) * C(S(x)) dx.
(2) C(x)^2 - S(x)^2 = 1.
(3a) d/dx S(x) = C(x) * C(S(x)).
(3b) d/dx C(x) = S(x) * C(S(x)).
(4a) C(x) + S(x) = exp( Integral C(S(x)) dx ).
(4b) C(x) = cosh( Integral C(S(x)) dx ).
(4c) S(x) = sinh( Integral C(S(x)) dx ).
(5) C(S(x))^2 - S(S(x))^2 = 1.
(5a) S(S(x)) = Integral C(x) * C(S(x))^2 * C(S(S(x))) dx.
(5b) C(S(x)) = 1 + Integral C(x) * S(S(x)) * C(S(x)) * C(S(S(x))) dx.
(6a) C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
(6b) C(S(x)) = cosh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
(6c) S(S(x)) = sinh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
(7) C(S(S(x))) + S(S(S(x))) = exp( Integral C(x) * C(S(x))^2 * C(S(S(x))) * C(S(S(S(x)))) dx ).
EXAMPLE
E.g.f. A(x) = C(x) + S(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 24*x^5/5! + 109*x^6/6! + 872*x^7/7! + 5737*x^8/8! + 67072*x^9/9! + 579961*x^10/10! + 9174400*x^11/11! + 98213933*x^12/12! + 1999010432*x^13/13! + 25474555941*x^14/14! + 644045742336*x^15/15! + 9505761607249*x^16/16! + 290850932891648*x^17/17! + 4872947687449969*x^18/18! + ...
such that C(x) + S(x) = exp( Integral C(S(x)) dx ).
RELATED SERIES.
S(x) = x + 2*x^3/3! + 24*x^5/5! + 872*x^7/7! + 67072*x^9/9! + 9174400*x^11/11! + 1999010432*x^13/13! + 644045742336*x^15/15! + 290850932891648*x^17/17! + ...
C(x) = 1 + x^2/2! + 5*x^4/4! + 109*x^6/6! + 5737*x^8/8! + 579961*x^10/10! + 98213933*x^12/12! + 25474555941*x^14/14! + 9505761607249*x^16/16! + 4872947687449969*x^18/18! + ...
C(S(x)) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 39929*x^8/8! + 5724249*x^10/10! + 1299323781*x^12/12! + 433635007877*x^14/14! + 201870080039537*x^16/16! + ...
S(S(x)) = x + 4*x^3/3! + 88*x^5/5! + 4992*x^7/7! + 549504*x^9/9! + 101239168*x^11/11! + 28464335360*x^13/13! + 11465663251456*x^15/15! + 6319308066455552*x^17/17! + ...
A(S(x)) = C(S(x)) + S(S(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4992*x^7/7! + 39929*x^8/8! + 549504*x^9/9! + 5724249*x^10/10! + 101239168*x^11/11! + 1299323781*x^12/12! + 28464335360*x^13/13! + 433635007877*x^14/14! + 11465663251456*x^15/15! + 201870080039537*x^16/16! + 6319308066455552*x^17/17! + ...
such that C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
PROG
(PARI) {a(n) = my(S=x, C=1); for(i=1, n,
S = intformal( C * subst(C, x, S) + x*O(x^(2*n)) );
C = 1 + intformal( S * subst(C, x, S) + x*O(x^(2*n)) ); );
n! * polcoeff( C+S, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 06 2019
STATUS
approved