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%I #3 Jan 06 2017 16:35:34
%S 1,1,1,1,5,21,101,581,3605,29141,239701,2205861,22832005,246573061,
%T 2990463301,38108222341,518783617205,7633121129781,116869543894901,
%U 1918479435194021,33025793008567205,595507639576003301,11356021765206934501,224730968057626784901,4667903267042610866005,101036430792914926153621,2269220040134583532958101
%N E.g.f. C(x) + S(x), where C(x)^2 - S(x)^2 = 1 and D(x)^3 - S(x)^3 = 1, and functions S(x), C(x), and D(x) are described by A280620, A280621, and A280622, respectively.
%F E.g.f. C(x) + S(x), where related functions S = S(x), C = C(x), and D = D(x) possess the following properties.
%F (1.a) C^2 - S^2 = 1.
%F (1.b) D^3 - S^3 = 1.
%F Integrals.
%F (2.a) S = Integral C*D^2 dx.
%F (2.b) C = 1 + Integral S*D^2 dx.
%F (2.c) D = 1 + Integral C*S^2 dx.
%F (2.d) C + S = 1 + Integral (C + S) * D^2 dx.
%F (2.e) D - S = 1 - Integral (D^2 - S^2) * C dx.
%F Exponential.
%F (3.a) C + S = exp( Integral D^2 dx ).
%F (3.b) D - S = exp( Integral -(D + S) * C dx.
%F (3.c) C = cosh( Integral D^2 dx ).
%F (3.d) S = sinh( Integral D^2 dx ).
%F Derivatives.
%F (4.a) S' = C*D^2.
%F (4.b) C' = S*D^2.
%F (4.c) D' = C*S^2.
%F (4.d) (C' + S')/(C + S) = D^2.
%F (4.e) (D' - S')/(D - S) = -(D + S) * C.
%e E.g.f.: C(x) + S(x) = 1 + x + x^2/2! + x^3/3! + 5*x^4/4! + 21*x^5/5! + 101*x^6/6! + 581*x^7/7! + 3605*x^8/8! + 29141*x^9/9! + 239701*x^10/10! + 2205861*x^11/11! + 22832005*x^12/12! + 246573061*x^13/13! + 2990463301*x^14/14! + 38108222341*x^15/15! + 518783617205*x^16/16! + 7633121129781*x^17/17! + 116869543894901*x^18/18! + 1918479435194021*x^19/19! + 33025793008567205*x^20/20! + 595507639576003301*x^21/21! +...
%e such that
%e (1) C(x) + S(x) = exp( Integral D(x)^2 dx ),
%e (2) C(x) + S(x) = 1 + Integral (C(x) + S(x)) * D(x)^2 dx,
%e where functions S(x), C(x), and D(x) are illustrated below.
%e RELATED SERIES.
%e S(x) = x + x^3/3! + 4*x^4/4! + x^5/5! + 100*x^6/6! + 161*x^7/7! + 1764*x^8/8! + 22001*x^9/9! + 49700*x^10/10! + 1649921*x^11/11! + 10057124*x^12/12! + 105372001*x^13/13! + 2044251300*x^14/14! + 12879413281*x^15/15! + 315936586084*x^16/16! + 3892292034001*x^17/17! + 49987743460900*x^18/18! +...
%e C(x) = 1 + x^2/2! + x^4/4! + 20*x^5/5! + x^6/6! + 420*x^7/7! + 1841*x^8/8! + 7140*x^9/9! + 190001*x^10/10! + 555940*x^11/11! + 12774881*x^12/12! + 141201060*x^13/13! + 946212001*x^14/14! + 25228809060*x^15/15! + 202847031121*x^16/16! + 3740829095780*x^17/17! + 66881800434001*x^18/18! +...
%e D(x) = 1 + 2*x^3/3! + 20*x^5/5! + 40*x^6/6! + 182*x^7/7! + 3360*x^8/8! + 5320*x^9/9! + 165480*x^10/10! + 1193962*x^11/11! + 7681520*x^12/12! + 182657020*x^13/13! + 1028347320*x^14/14! + 21430373342*x^15/15! + 296385660480*x^16/16! + 2926954283120*x^17/17! + 74104327031560*x^18/18! +...
%e S(x)^2 = 2*x^2/2! + 8*x^4/4! + 40*x^5/5! + 32*x^6/6! + 1680*x^7/7! + 3808*x^8/8! + 49560*x^9/9! + 646912*x^10/10! + 2192960*x^11/11! + 65759008*x^12/12! + 475555080*x^13/13! + 5786067392*x^14/14! + 114473289840*x^15/15! + 891694992608*x^16/16! + 21934824868600*x^17/17! + 298444830841472*x^18/18! +...
%e such that C(x)^2 = 1 + S(x)^2.
%e D(x)^2 = 1 + 4*x^3/3! + 40*x^5/5! + 160*x^6/6! + 364*x^7/7! + 11200*x^8/8! + 24080*x^9/9! + 519120*x^10/10! + 5344724*x^11/11! + 27288800*x^12/12! + 752580920*x^13/13! + 5142016880*x^14/14! + 86718961084*x^15/15! + 1483995676800*x^16/16! + 13774998062560*x^17/17! + 356032443815440*x^18/18! +...
%e such that D(x)^2 = S'(x)/C(x) = C'(x)/S(x).
%e S(x)^3 = 6*x^3/3! + 60*x^5/5! + 360*x^6/6! + 546*x^7/7! + 23520*x^8/8! + 69720*x^9/9! + 1060920*x^10/10! + 14669886*x^11/11! + 67692240*x^12/12! + 1957699380*x^13/13! + 16377040680*x^14/14! + 228086752026*x^15/15! + 4642872212160*x^16/16! + 43205148425040*x^17/17! + 1084693228559640*x^18/18! +...
%e such that D(x)^3 = 1 + S(x)^3.
%e 1/(D(x) - S(x)) = 1 + x + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 81*x^5/5! + 452*x^6/6! + 2765*x^7/7! + 19460*x^8/8! + 156121*x^9/9! + 1368052*x^10/10! + 13327125*x^11/11! + 141326500*x^12/12! + 1616350561*x^13/13! + 20040895252*x^14/14! + 264759181085*x^15/15! + 3740415315140*x^16/16! + 56164918735401*x^17/17! + 891038080096052*x^18/18! + 14957788277468645*x^19/19! + 263869908657105380*x^20/20! + 4889789934063374641*x^21/21! +...
%e such that 1/(D(x) - S(x)) = exp( Integral (D(x) + S(x)) * C(x) dx.
%o (PARI) {a(n) = my(S=x,C=1,D=1); for(i=0,n, S = intformal( C*D^2 + x*O(x^n)); C = 1 + intformal( S*D^2 ); D = 1 + intformal( C*S^2 )); n!*polcoeff(C+S,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A280620 (S), A280621 (C), A280622 (D), A280624 (1/(C-S)).
%K nonn
%O 0,5
%A _Paul D. Hanna_, Jan 06 2017
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