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A292181
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E.g.f. A(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that A'(x) = A(x) + B(x)*C(x).
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3
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1, 3, 10, 45, 259, 1806, 14827, 140367, 1504576, 17972559, 236275711, 3387012720, 52572376669, 878552787927, 15729439074058, 300400031036745, 6095885898471775, 130982551821899862, 2970844882925223487, 70929401617621416243, 1778125633605205346584, 46698342082602696345555, 1282168260097348871508667, 36734284970419645262875200, 1096293296048734274708523433, 34026339905854090378353208155
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OFFSET
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1,2
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COMMENTS
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Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).
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LINKS
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FORMULA
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E.g.f. A(x) and related functions B(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
(5a) B(x) + i*A(x) = C(x) * exp( i*atan( A(x)/B(x) ) ).
(5b) A(x)/B(x) = Series_Reversion( Integral 1/( sqrt(1-x^4) * (1 + Integral 1/sqrt(1-x^4) dx) ) dx ).
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EXAMPLE
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E.g.f.: A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
RELATED SERIES.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).
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PROG
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(PARI) {a(n) = my(A=x, B=1, C=1); for(i=0, n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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