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 A372814 #16 May 17 2024 10:20:09
%S A372814 1,7,381,50051,11899705,4787171775,2800735142453,2286983798222779,
%T A372814 2476757127978318705,3434360322639603491447,5940446259665147492879341,
%U A372814 12533181362722474751715110643,31687559294370295303515685200041,94578054008984518849163257005668911
%N A372814 Expansion of e.g.f. T(x) satisfying T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ), where a(n) is the coefficient of x^(2*n+1)/(2*n+1)! in T(x) for n >= 0.
%H A372814 Paul D. Hanna, <a href="/A372814/b372814.txt">Table of n, a(n) for n = 0..299</a>
%F A372814 a(n) = Sum_{j=0..n} A370433(n,j) * 2^(2*j).
%F A372814 E.g.f.: T(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
%F A372814 Definition.
%F A372814 (1.a) (C + S) = exp(x*D).
%F A372814 (1.b) (D + 2*T) = exp(2*x*C).
%F A372814 (2.a) C^2 - S^2 = 1.
%F A372814 (2.b) D^2 - 4*T^2 = 1.
%F A372814 Hyperbolic functions.
%F A372814 (3.a) C = cosh(x*D).
%F A372814 (3.b) S = sinh(x*D).
%F A372814 (3.c) D = cosh(2*x*C).
%F A372814 (3.d) T = (1/2) * sinh(2*x*C).
%F A372814 (4.a) C = cosh( x*cosh(2*x*C) ).
%F A372814 (4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
%F A372814 (4.c) D = cosh( 2*x*cosh(x*D) ).
%F A372814 (4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
%F A372814 (5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
%F A372814 (5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
%F A372814 Integrals.
%F A372814 (6.a) C = 1 + Integral S*D + x*S*D' dx.
%F A372814 (6.b) S = Integral C*D + x*C*D' dx.
%F A372814 (6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
%F A372814 (6.d) T = Integral D*C + x*D*C' dx.
%F A372814 Derivatives (d/dx).
%F A372814 (7.a) C*C' = S*S'.
%F A372814 (7.b) D*D' = 4*T*T'.
%F A372814 (8.a) C' = S * (D + x*D').
%F A372814 (8.b) S' = C * (D + x*D').
%F A372814 (8.c) D' = 4 * T * (C + x*C').
%F A372814 (8.d) T' = D * (C + x*C').
%F A372814 (9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372814 (9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372814 (9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372814 (9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372814 (10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372814 (10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372814 Logarithms.
%F A372814 (11.a) D = log(C + sqrt(C^2 - 1)) / x.
%F A372814 (11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
%F A372814 (11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
%F A372814 (11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
%F A372814 The radius of convergence r of e.g.f. T(x) is r = 0.458693345589772637742719473602361341151810356245785213... where T(r) = 0.989147448863398861152401518907145018698758995598697027...
%e A372814 E.g.f: T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
%e A372814 and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
%e A372814 RELATED SERIES.
%e A372814 Related functions C(x), S(x), and D(x) are described below.
%e A372814 C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
%e A372814 where C(x) = cosh( x*sqrt(1 + 4*T(x)^2) )
%e A372814 and C(x) = cosh( x*cosh(2*x*C(x)) ).
%e A372814 S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
%e A372814 where S(x) = sinh( x*sqrt(1 + 4*T(x)^2) )
%e A372814 and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
%e A372814 D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
%e A372814 where D(x) = sqrt(1 + 4*T(x)^2)
%e A372814 and D(x) = cosh( 2*x*cosh(x*D(x)) ).
%e A372814 SPECIFIC VALUES.
%e A372814 T(1/3) = 0.396995956737823895057073833881324565170496359298875629...
%e A372814 T(1/4) = 0.272098438758403062037693234896743179247292418834110582...
%e A372814 T(1/5) = 0.210496129492378206205507948355083031094691532837329579...
%e A372814 T(1/6) = 0.172515307079869202329210004732936197729646250794708377...
%e A372814 T(1/10) = 0.101199443780546489307264980334019941975165460819500909...
%o A372814 (PARI) /* From T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ) */
%o A372814 {a(n) = my(T=x); for(i=0,n, T=truncate(T); T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2 + x*O(x^(2*i)) )) ));
%o A372814 (2*n+1)! * polcoeff(T, 2*n+1, x)}
%o A372814 for(n=0, 30, print1( a(n), ", "))
%o A372814 (PARI) /* From A370433 at k = 2 */
%o A372814 {a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
%o A372814 for(i=1, 2*n,
%o A372814 C = cosh( x*cosh(k*x*C +Ox) );
%o A372814 S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
%o A372814 D = cosh( k*x*cosh(x*D +Ox));
%o A372814 T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
%o A372814 (2*n+1)! * polcoeff(T, 2*n+1, x)}
%o A372814 for(n=0, 30, print1( a(n), ", "))
%Y A372814 Cf. A370433 (k = 2), A372811 (C(x)), A372812 (S(x)), A372813 (D(x)), A007106.
%K A372814 nonn
%O A372814 0,2
%A A372814 _Paul D. Hanna_, May 16 2024