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%I #11 Nov 28 2016 09:24:50
%S 1,1,1,3,9,39,189,1137,7521,58221,487521,4615623,46747449,523484019,
%T 6218441469,80413567317,1095843999681,16072230046041,247107215918241,
%U 4053246141598443,69395454770712489,1258826280827924799,23749475226740969949,472083799922946212697,9730211267060692468641,210327336751547848824261,4701988645468367963255361,109812853605044722106919663
%N E.g.f. C(x) + S(x), such that C(x)^2 - S(x)^2 = 1, 3*C(x)^2 - 2*D(x)^3 = 1, and D(x) = 1 + Integral S(x)*C(x) dx.
%H Paul D. Hanna, <a href="/A278749/b278749.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. A(x) = C(x) + S(x), where S(x), C(x), and D(x) satisfy:
%F (1) A(x) = exp( Integral D(x)^2 dx ),
%F (2) A(x) = 1 + Integral A(x)*D(x)^2 dx,
%F (3) S(x) = Integral C(x)*D(x)^2 dx,
%F (4) C(x) = 1 + Integral S(x)*D(x)^2 dx,
%F (5) D(x) = 1 + Integral S(x)*C(x) dx,
%F (6) C(x)^2 - S(x)^2 = 1,
%F (7) 3*C(x)^2 - 2*D(x)^3 = 1,
%F (8) 2*D(x)^3 - 3*S(x)^2 = 2.
%e E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 39*x^5/5! + 189*x^6/6! + 1137*x^7/7! + 7521*x^8/8! + 58221*x^9/9! + 487521*x^10/10! + 4615623*x^11/11! + 46747449*x^12/12! +...
%e such that A(x) = C(x) + S(x) where
%e 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! +...
%e 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! +...
%e Related expansions
%e 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! +...
%e 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! +...
%e 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! +...
%e 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! +...
%e satisfy
%e C(x)^2 - S(x)^2 = 1,
%e 3*C(x)^2 - 2*D(x)^3 = 1.
%e Logarithm of the e.g.f. begins:
%e log(A(x)) = x + 2*x^3/3! + 18*x^5/5! + 408*x^7/7! + 17352*x^9/9! + 1184832*x^11/11! + 118618128*x^13/13! + 16371203328*x^15/15! +...
%e which equals Integral D(x)^2 dx.
%o (PARI) {a(n) = my(S=x, C=1, D=1); for(i=1,n, S = intformal(C*(D^2 +O(x^(n+1)))); C = 1 + intformal(S*(D^2 +O(x^(n+1)))); D = 1 + intformal(S*C); ); n!*polcoeff(C+S,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A278750 (S(x)), A278751 (C(x)), A278752 (D(x)).
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 27 2016
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