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 A245166 #10 Feb 26 2025 02:34:26
%S A245166 1,5,25,215,2425,33875,569125,11160035,250047025,6302723075,
%T A245166 176522216125,5438291613155,182773714292425,6654680279353475,
%U A245166 260930805319957525,10961922511422743075,491220886240696086625,23388149451193115459075,1179066988050425638569325
%N A245166 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (cosh(2*x) - sinh(2*x)*cosh(3*x)).
%C A245166 Limit (a(n)/n!)^(-1/n) = log(t) = 0.3570506972213... where t satisfies 1 + t^2 + t^3 = t^5.
%F A245166 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (1 - sinh(2*x)^2*sinh(3*x)^2).
%F A245166 E.g.f.: (cosh(2*x)*cosh(3*x) + sinh(2*x) + sinh(3*x)) / (1 - sinh(2*x)*sinh(3*x)). - _Paul D. Hanna_, Dec 22 2018
%F A245166 E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245164 and A245165, respectively.
%F A245166 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 A245166 (1) A0(x)^2 - A1(x)^2 = 1.
%F A245166 (2) exp(2*x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).
%F A245166 (3) exp(3*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 A245166 From _Paul D. Hanna_, Dec 22 2018: (Start)
%F A245166 (4) exp(2*x) = (A0(x)*cosh(3*x) + A1(x) - sinh(3*x)) / (1 + sinh(3*x)*A1(x)).
%F A245166 (5) exp(3*x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)). (End)
%F A245166 FORMULAS FOR TERMS.
%F A245166 a(n) == 0 (mod 5) for n>0.
%F A245166 From _Paul D. Hanna_, Dec 22 2018: (Start)
%F A245166 a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * A322620(n,k).
%F A245166 a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * A322190(n,k). (End)
%e A245166 E.g.f.: A(x) = 1 + 5*x + 25*x^2/2! + 215*x^3/3! + 2425*x^4/4! + 33875*x^5/5! +...
%e A245166 such that A(x) = B(x)*C(x), where
%e A245166 B(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
%e A245166 C(x) = 1 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
%e A245166 are the e.g.f.s of A245164 and A245165, respectively.
%e A245166 Let A(x) = A0(x) + A1(x) where
%e A245166 A0(x) = 1 + 25*x^2/2! + 2425*x^4/4! + 569125*x^6/6! + 250047025*x^8/8! +...
%e A245166 A1(x) = 5*x + 215*x^3/3! + 33875*x^5/5! + 11160035*x^7/7! + 6302723075*x^9/9! +...
%e A245166 then A0(x)^2 - A1(x)^2 = 1.
%e A245166 Note that the logarithm is an odd function:
%e A245166 log(A(x)) = 5*x + 90*x^3/3! + 8250*x^5/5! + 1946910*x^7/7! + 855018450*x^9/9! + 603621345030*x^11/11! + 624997732481850*x^13/13! +...
%e A245166 thus A(x)*A(-x) = 1.
%t A245166 With[{nn=20},CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/(Cosh[2x]-Sinh[2x]Cosh[3x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 21 2024 *)
%o A245166 (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (cosh(2*X) - sinh(2*X)*cosh(3*X)), n)}
%o A245166 for(n=0, 30, print1(a(n), ", "))
%o A245166 (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (1 - sinh(2*X)^2*sinh(3*X)^2), n)}
%o A245166 for(n=0, 30, print1(a(n), ", "))
%Y A245166 Cf. A245164, A245165, A245140, A245155.
%Y A245166 Cf. A322620, A322190.
%K A245166 nonn
%O A245166 0,2
%A A245166 _Paul D. Hanna_, Jul 12 2014