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 A372812 #15 May 17 2024 10:19:59
%S A372812 1,13,441,68069,15591025,6212017725,3652639410473,2963960104898581,
%T A372812 3208843075117716705,4442917542274682028653,7676236962804930027455641,
%U A372812 16182752346241750118582151237,40883629770018829153233694565201,121951983267795526035606825074967709
%N A372812 Expansion of e.g.f. S(x) satisfying S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ), where a(n) is the coefficient of x^(2*n+1)/(2*n+1)! in S(x) for n >= 0.
%H A372812 Paul D. Hanna, <a href="/A372812/b372812.txt">Table of n, a(n) for n = 0..300</a>
%F A372812 a(n) = Sum_{j=0..n} A370431(n,j) * 2^(2*j).
%F A372812 E.g.f.: S(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 A372812 Definition.
%F A372812 (1.a) (C + S) = exp(x*D).
%F A372812 (1.b) (D + 2*T) = exp(2*x*C).
%F A372812 (2.a) C^2 - S^2 = 1.
%F A372812 (2.b) D^2 - 4*T^2 = 1.
%F A372812 Hyperbolic functions.
%F A372812 (3.a) C = cosh(x*D).
%F A372812 (3.b) S = sinh(x*D).
%F A372812 (3.c) D = cosh(2*x*C).
%F A372812 (3.d) T = (1/2) * sinh(2*x*C).
%F A372812 (4.a) C = cosh( x*cosh(2*x*C) ).
%F A372812 (4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
%F A372812 (4.c) D = cosh( 2*x*cosh(x*D) ).
%F A372812 (4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
%F A372812 (5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
%F A372812 (5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
%F A372812 Integrals.
%F A372812 (6.a) C = 1 + Integral S*D + x*S*D' dx.
%F A372812 (6.b) S = Integral C*D + x*C*D' dx.
%F A372812 (6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
%F A372812 (6.d) T = Integral D*C + x*D*C' dx.
%F A372812 Derivatives (d/dx).
%F A372812 (7.a) C*C' = S*S'.
%F A372812 (7.b) D*D' = 4*T*T'.
%F A372812 (8.a) C' = S * (D + x*D').
%F A372812 (8.b) S' = C * (D + x*D').
%F A372812 (8.c) D' = 4 * T * (C + x*C').
%F A372812 (8.d) T' = D * (C + x*C').
%F A372812 (9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372812 (9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372812 (9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372812 (9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372812 (10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372812 (10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372812 Logarithms.
%F A372812 (11.a) D = log(C + sqrt(C^2 - 1)) / x.
%F A372812 (11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
%F A372812 (11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
%F A372812 (11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
%F A372812 The radius of convergence r of e.g.f. S(x) is r = 0.458693345589772637742719473602361341151810356245785213... where S(r) = 1.201251917668278563521948977625996579820943724944393208...
%e A372812 E.g.f: 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 A372812 and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
%e A372812 RELATED SERIES.
%e A372812 Related functions C(x), D(x), and T(x) are described below.
%e A372812 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 A372812 where C(x) = sqrt(1 + S(x)^2)
%e A372812 and C(x) = cosh( x*cosh(2*x*C(x)) ).
%e A372812 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 A372812 where D(x) = cosh( 2*x*sqrt(1 + S(x)^2) )
%e A372812 and D(x) = cosh( 2*x*cosh(x*D(x)) ).
%e A372812 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 A372812 where T(x) = (1/2) * sinh( 2*x*sqrt(1 + S(x)^2) )
%e A372812 and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
%e A372812 SPECIFIC VALUES.
%e A372812 S(1/3) = 0.438594611804336870818029761992727975330083659221250216...
%e A372812 S(1/4) = 0.288479916487512228329919975913022787931012140199922189...
%e A372812 S(1/5) = 0.218707961000324022488369693038572482223647706535551198...
%e A372812 S(1/6) = 0.177223127385698497600070746700827976044841583345600952...
%e A372812 S(1/10) = 0.102204811824008710495811173453365253815203645781101342...
%o A372812 (PARI) /* From S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ) */
%o A372812 {a(n) = my(S=x); for(i=0,n, S=truncate(S); S = sinh( x*cosh(2*x*sqrt(1 + S^2 + x*O(x^(2*i)) )) ));
%o A372812 (2*n+1)! * polcoeff(S, 2*n+1, x)}
%o A372812 for(n=0, 30, print1( a(n), ", "))
%o A372812 (PARI) /* From A370431 at k = 2 */
%o A372812 {a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
%o A372812 for(i=1, 2*n,
%o A372812 C = cosh( x*cosh(k*x*C +Ox) );
%o A372812 S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
%o A372812 D = cosh( k*x*cosh(x*D +Ox));
%o A372812 T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
%o A372812 (2*n+1)! * polcoeff(S, 2*n+1, x)}
%o A372812 for(n=0, 30, print1( a(n), ", "))
%Y A372812 Cf. A370431 (k = 2), A372811 (C(x)), A372813 (D(x)), A372814 (T(x)), A007106.
%K A372812 nonn
%O A372812 0,2
%A A372812 _Paul D. Hanna_, May 16 2024