OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. S(x) and related series C(x) and D(x) satisfy:
(1) S(x) = Integral C(x)*D(x)^2 dx,
(2) C(x) = 1 + Integral S(x)*D(x)^2 dx,
(3) D(x) = 1 + Integral S(x)*C(x) dx,
(4) C(x)^2 - S(x)^2 = 1,
(5) 3*C(x)^2 - 2*D(x)^3 = 1,
(6) 2*D(x)^3 - 3*S(x)^2 = 2,
(7) C(x) + S(x) = exp( Integral D(x)^2 dx ).
EXAMPLE
E.g.f.: S(x) = x + 3*x^3/3! + 39*x^5/5! + 1137*x^7/7! + 58221*x^9/9! + 4615623*x^11/11! + 523484019*x^13/13! + 80413567317*x^15/15! + 16072230046041*x^17/17! + 4053246141598443*x^19/19! +...
and related series
C(x) = 1 + x^2/2! + 9*x^4/4! + 189*x^6/6! + 7521*x^8/8! + 487521*x^10/10! + 46747449*x^12/12! + 6218441469*x^14/14! + 1095843999681*x^16/16! + 247107215918241*x^18/18! +...
D(x) = 1 + x^2/2! + 6*x^4/4! + 114*x^6/6! + 4224*x^8/8! + 258696*x^10/10! + 23685696*x^12/12! + 3030422544*x^14/14! + 516368179584*x^16/16! + 113039478326016*x^18/18! +...
satisfy
C(x)^2 - S(x)^2 = 1,
3*C(x)^2 - 2*D(x)^3 = 1.
Related expansions.
C(x)^2 = 1 + 2*x^2/2! + 24*x^4/4! + 648*x^6/6! + 31296*x^8/8! + 2366352*x^10/10! + 257865984*x^12/12! + 38266414848*x^14/14! + 7419295374336*x^16/16! + 1820980419409152*x^18/18! +...
D(x)^2 = 1 + 2*x^2/2! + 18*x^4/4! + 408*x^6/6! + 17352*x^8/8! + 1184832*x^10/10! + 118618128*x^12/12! + 16371203328*x^14/14! + 2979295540992*x^16/16! + 691248148134912*x^18/18! +...
D(x)^3 = 1 + 3*x^2/2! + 36*x^4/4! + 972*x^6/6! + 46944*x^8/8! + 3549528*x^10/10! + 386798976*x^12/12! + 57399622272*x^14/14! + 11128943061504*x^16/16! + 2731470629113728*x^18/18! +...
such that 2*D(x)^3 - 3*S(x)^2 = 2.
PROG
(PARI) {a(n) = my(S=x, C=1, D=1); for(i=1, 2*n, S = intformal(C*(D^2 +O(x^(2*n+2)))); C = 1 + intformal(S*(D^2 +O(x^(2*n+2)))); D = 1 + intformal(S*C); ); (2*n+1)!*polcoeff(S, 2*n+1)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 27 2016
STATUS
approved