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 A245140 #30 Feb 26 2025 02:34:14
%S A245140 1,3,9,45,297,2433,23949,275145,3612177,53348193,875453589,
%T A245140 15802999545,311196040857,6638817262353,152521855979229,
%U A245140 3754366520240745,98575724288354337,2749997026637342913,81230299711952152869,2532707187355266614745,83124358113443446120617,2864579803637260793877873
%N A245140 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(2*x)).
%C A245140 Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.
%C A245140 In general, the radius of convergence r of the e.g.f. (cosh(p*x) + sinh(p*x)*cosh(q*x)) / (cosh(q*x) - sinh(q*x)*cosh(p*x)), where p and q are positive integers, equals r = log(t) such that t is the positive real root that satisfies: 1 + t^p + t^q = t^(p+q).
%F A245140 E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) * (cosh(2*x) + sinh(2*x)*cosh(x)) / (1 - sinh(x)^2*sinh(2*x)^2).
%F A245140 E.g.f.: (cosh(x)*cosh(2*x) + sinh(x) + sinh(2*x)) / (1 - sinh(x)*sinh(2*x)). - _Paul D. Hanna_, Dec 22 2018
%F A245140 E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245138 and A245139, respectively.
%F A245140 Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
%F A245140 (1) A0(x)^2 - A1(x)^2 = 1.
%F A245140 (2) exp(x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).
%F A245140 (3) exp(2*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).
%F A245140 From _Paul D. Hanna_, Dec 22 2018: (Start)
%F A245140 (4) exp(x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)).
%F A245140 (5) exp(2*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / (1 + sinh(x)*A1(x)). (End)
%F A245140 FORMULAS FOR TERMS.
%F A245140 From _Paul D. Hanna_, Dec 22 2018: (Start)
%F A245140 a(n) = Sum_{k=0..n} 2^k * A322620(n,k).
%F A245140 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * A322190(n,k). (End)
%e A245140 E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 45*x^3/3! + 297*x^4/4! + 2433*x^5/5! +...
%e A245140 such that A(x) = B(x)*C(x), where
%e A245140 B(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! + 3121*x^6/6! +...
%e A245140 C(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! + 5344*x^6/6! +...
%e A245140 are the e.g.f.s of A245138 and A245139, respectively.
%e A245140 Let A(x) = A0(x) + A1(x) where
%e A245140 A0(x) = 1 + 9*x^2/2! + 297*x^4/4! + 23949*x^6/6! + 3612177*x^8/8! +...
%e A245140 A1(x) = 3*x + 45*x^3/3! + 2433*x^5/5! + 275145*x^7/7! + 53348193*x^9/9! +...
%e A245140 then A0(x)^2 - A1(x)^2 = 1.
%e A245140 Note that the logarithm is an odd function:
%e A245140 log(A(x)) = 3*x + 18*x^3/3! + 570*x^5/5! + 46158*x^7/7! + 6959250*x^9/9! + 1686709398*x^11/11! + 599570355930*x^13/13! + 3754366520240745*x^15/15! +...
%e A245140 thus A(x)*A(-x) = 1.
%o A245140 (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(2*X) + sinh(2*X)*cosh(X)) / (cosh(X) - sinh(X)*cosh(2*X)), n)}
%o A245140 for(n=0,30,print1(a(n),", "))
%o A245140 (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(2*X)) * (cosh(2*X) + sinh(2*X)*cosh(X)) / (1 - sinh(X)^2*sinh(2*X)^2),n)}
%o A245140 for(n=0,30,print1(a(n),", "))
%Y A245140 Cf. A245138, A245139, A245155, A245166.
%Y A245140 Cf. A322620, A322190.
%K A245140 nonn
%O A245140 0,2
%A A245140 _Paul D. Hanna_, Jul 12 2014