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%I #24 Sep 11 2017 13:30:20
%S 1,1,3,13,63,361,2499,20581,196311,2116561,25357563,333765037,
%T 4787007855,74323701817,1242253733619,22243082373301,424815246293319,
%U 8620744969300321,185235767397027627,4201390722798810493,100309092062158564959,2514646421630798317897,66041388198395188082595,1813259146315114344920581,51950114633383773360554679,1550392693763071812557794801,48120508780248064233484223067
%N 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).
%C Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
%C 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).
%H Paul D. Hanna, <a href="/A292183/b292183.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f. C(x) and related functions A(x) and B(x) satisfy:
%F (1a) A(x)^2 + B(x)^2 = C(x)^2.
%F (1b) B(x)^2 - A(x)^2 = exp(x)^2.
%F (1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
%F (2a) A(x) = Integral A(x) + B(x)*C(x) dx.
%F (2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
%F (2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
%F (3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
%F (3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
%F (3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
%F (3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
%F (4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
%F (4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
%F (4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
%F Limit A292183(n)/A292181(n) = sqrt(2).
%F Limit A292183(n)/A292182(n) = sqrt(2).
%e 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! +...
%e where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
%e RELATED SERIES.
%e 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! +...
%e where A(x) = Integral A(x) + B(x)*C(x) dx.
%e 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 +...
%e where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
%e Squares of series.
%e 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! +...
%e where A(x)^2 + B(x)^2 = C(x)^2.
%e 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! +...
%e where B(x)^2 - A(x)^2 = exp(2*x).
%e 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! +...
%e where C(x)^2 - 2*A(x)^2 = exp(2*x).
%o (PARI) {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
%o B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(C,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A292181 (A), A292182 (B).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 10 2017
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