This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A292182 #23 Sep 11 2017 13:29:59
%S A292182 1,1,2,7,35,226,1715,14701,141248,1515661,18048527,236581984,
%T A292182 3386091821,52533799501,877993866290,15723411375931,300349139257727,
%U A292182 6095613429234730,130983518612114231,2970900143887175977,70930381205350706888,1778137090832694851161,46698407537794612100459,1282167191852237842607584,36734238381564939631425737,1096292258727541156091352361,34026322932421876848090674594
%N A292182 E.g.f. B(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that B'(x) = B(x) + A(x)*C(x).
%C A292182 Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
%C A292182 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 A292182 Paul D. Hanna, <a href="/A292182/b292182.txt">Table of n, a(n) for n = 0..300</a>
%F A292182 E.g.f. B(x) and related functions A(x) and C(x) satisfy:
%F A292182 (1a) A(x)^2 + B(x)^2 = C(x)^2.
%F A292182 (1b) B(x)^2 - A(x)^2 = exp(x)^2.
%F A292182 (1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
%F A292182 (2a) A(x) = Integral A(x) + B(x)*C(x) dx.
%F A292182 (2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
%F A292182 (2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
%F A292182 (3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
%F A292182 (3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
%F A292182 (3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
%F A292182 (3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
%F A292182 (4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
%F A292182 (4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
%F A292182 (4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
%F A292182 Limit A292181(n)/A292182(n) = 1.
%F A292182 Limit A292183(n)/A292182(n) = sqrt(2).
%e A292182 E.g.f.: 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 A292182 where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
%e A292182 RELATED SERIES.
%e A292182 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 A292182 where A(x) = Integral A(x) + B(x)*C(x) dx.
%e A292182 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 A292182 where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
%e A292182 Squares of series.
%e A292182 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 A292182 where A(x)^2 + B(x)^2 = C(x)^2.
%e A292182 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 A292182 where B(x)^2 - A(x)^2 = exp(2*x).
%e A292182 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 A292182 where C(x)^2 - 2*A(x)^2 = exp(2*x).
%o A292182 (PARI) {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
%o A292182 B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(B,n)}
%o A292182 for(n=0,30,print1(a(n),", "))
%Y A292182 Cf. A292181 (A), A292183 (C).
%K A292182 nonn
%O A292182 0,3
%A A292182 _Paul D. Hanna_, Sep 10 2017