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A292183
E.g.f. C(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that C'(x) = C(x) + 2*A(x)*B(x).
3
1, 1, 3, 13, 63, 361, 2499, 20581, 196311, 2116561, 25357563, 333765037, 4787007855, 74323701817, 1242253733619, 22243082373301, 424815246293319, 8620744969300321, 185235767397027627, 4201390722798810493, 100309092062158564959, 2514646421630798317897, 66041388198395188082595, 1813259146315114344920581, 51950114633383773360554679, 1550392693763071812557794801, 48120508780248064233484223067
OFFSET
0,3
COMMENTS
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).
LINKS
FORMULA
E.g.f. C(x) and related functions A(x) and B(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 ).
Limit A292183(n)/A292181(n) = sqrt(2).
Limit A292183(n)/A292182(n) = sqrt(2).
EXAMPLE
E.g.f.: 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.
RELATED SERIES.
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.
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.
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).
PROG
(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(C, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A292181 (A), A292182 (B).
Sequence in context: A202837 A370396 A180111 * A006923 A349304 A372507
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 10 2017
STATUS
approved