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%I #11 Nov 30 2016 21:31:56
%S 1,1,6,114,4224,258696,23685696,3030422544,516368179584,
%T 113039478326016,30915842271630336,10330366155858849024,
%U 4141017299122378758144,1961342355370645525671936,1083606291089708175858917376,690681085734140756895484053504,503068200949361929673857570504704,415234815803178624028164344747360256,385549194671700625768876635402899030016
%N E.g.f. D(x) = 1 + Integral S(x)*C(x) dx, where C(x)^2 - S(x)^2 = 1 and 3*C(x)^2 - 2*D(x)^3 = 1.
%H Paul D. Hanna, <a href="/A278752/b278752.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. D(x) and related series S(x) and C(x) satisfy:
%F (1) S(x) = Integral C(x)*D(x)^2 dx,
%F (2) C(x) = 1 + Integral S(x)*D(x)^2 dx,
%F (3) D(x) = 1 + Integral S(x)*C(x) dx,
%F (4) C(x)^2 - S(x)^2 = 1,
%F (5) 3*C(x)^2 - 2*D(x)^3 = 1,
%F (6) 2*D(x)^3 - 3*S(x)^2 = 2,
%F (7) C(x) + S(x) = exp( Integral D(x)^2 dx ).
%e E.g.f.: 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 and related series
%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 satisfy
%e C(x)^2 - S(x)^2 = 1,
%e 3*C(x)^2 - 2*D(x)^3 = 1.
%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)^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 such that 2*D(x)^3 - 3*S(x)^2 = 2.
%o (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)!*polcoeff(D,2*n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A278750 (S(x)), A278751 (C(x)), A278749 (C(x) + S(x)).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 27 2016
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