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 A372811 #14 May 17 2024 10:20:14
%S A372811 1,1,49,3601,680737,218915041,105958624465,74506995584113,
%T A372811 70436550855565633,86815671664245679297,135090335333407225545841,
%U A372811 258969022695032433287216593,599973069857987759584855153249,1652347283935245955005795151113121,5336236426918250608377414155884578577
%N A372811 Expansion of e.g.f. C(x) satisfying C(x) = cosh( x*cosh(2*x*C(x)) ), where a(n) is the coefficient of x^(2*n)/(2*n)! in C(x) for n >= 0.
%H A372811 Paul D. Hanna, <a href="/A372811/b372811.txt">Table of n, a(n) for n = 0..300</a>
%F A372811 a(n) = Sum_{j=0..n} A370430(n,j) * 2^(2*j).
%F A372811 E.g.f.: C(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
%F A372811 Definition.
%F A372811 (1.a) (C + S) = exp(x*D).
%F A372811 (1.b) (D + 2*T) = exp(2*x*C).
%F A372811 (2.a) C^2 - S^2 = 1.
%F A372811 (2.b) D^2 - 4*T^2 = 1.
%F A372811 Hyperbolic functions.
%F A372811 (3.a) C = cosh(x*D).
%F A372811 (3.b) S = sinh(x*D).
%F A372811 (3.c) D = cosh(2*x*C).
%F A372811 (3.d) T = (1/2) * sinh(2*x*C).
%F A372811 (4.a) C = cosh( x*cosh(2*x*C) ).
%F A372811 (4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
%F A372811 (4.c) D = cosh( 2*x*cosh(x*D) ).
%F A372811 (4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
%F A372811 (5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
%F A372811 (5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
%F A372811 Integrals.
%F A372811 (6.a) C = 1 + Integral S*D + x*S*D' dx.
%F A372811 (6.b) S = Integral C*D + x*C*D' dx.
%F A372811 (6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
%F A372811 (6.d) T = Integral D*C + x*D*C' dx.
%F A372811 Derivatives (d/dx).
%F A372811 (7.a) C*C' = S*S'.
%F A372811 (7.b) D*D' = 4*T*T'.
%F A372811 (8.a) C' = S * (D + x*D').
%F A372811 (8.b) S' = C * (D + x*D').
%F A372811 (8.c) D' = 4 * T * (C + x*C').
%F A372811 (8.d) T' = D * (C + x*C').
%F A372811 (9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372811 (9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372811 (9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372811 (9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372811 (10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
%F A372811 (10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
%F A372811 Logarithms.
%F A372811 (11.a) D = log(C + sqrt(C^2 - 1)) / x.
%F A372811 (11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
%F A372811 (11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
%F A372811 (11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
%F A372811 The radius of convergence r and C(r) satisfy r = arccosh(C(r)) / cosh(2*r*C(r)) and 1 = 2*r^2 * sinh(2*r*C(r)) * sinh( r*cosh(2*r*C(r)) ), where r = 0.458693345589772637742719473602361341151810356245785213... and C(r) = 1.56301189045436141892741676499724550339640443730107995...
%e A372811 E.g.f: 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 A372811 where C(x) = cosh( x*cosh(2*x*C(x)) ).
%e A372811 RELATED SERIES.
%e A372811 Related functions S(x), D(x), and T(x) are described below.
%e A372811 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 A372811 where S(x) = sqrt(C(x)^2 - 1)
%e A372811 and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
%e A372811 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 A372811 where D(x) = cosh( 2*x*C(x) )
%e A372811 and D(x) = cosh( 2*x*cosh(x*D(x)) ).
%e A372811 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 A372811 where T(x) = (1/2) * sinh( 2*x*C(x) )
%e A372811 and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
%e A372811 SPECIFIC VALUES.
%e A372811 C(1/3) = 1.09195477630888952286167165173829275218422327069159600...
%e A372811 C(1/4) = 1.04077887287196699205886551058236704630676681245696946...
%e A372811 C(1/5) = 1.02363722685574465853941118194990482596731360834136139...
%e A372811 C(1/6) = 1.01558260957952973250484327981285267794556192126351939...
%e A372811 C(1/10) = 1.00520934315195311495083109289140774006817550223233669...
%o A372811 (PARI) /* From C(x) = cosh( x*cosh(2*x*C(x)) ) */
%o A372811 {a(n) = my(C=1); for(i=0,n, C=truncate(C); C = cosh( x*cosh(2*x*C + x*O(x^(2*i))) ));
%o A372811 (2*n)! * polcoeff(C, 2*n, x)}
%o A372811 for(n=0, 30, print1( a(n), ", "))
%o A372811 (PARI) /* From A370430 at k = 2 */
%o A372811 {a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
%o A372811 for(i=1, 2*n,
%o A372811 C = cosh( x*cosh(k*x*C +Ox) );
%o A372811 S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
%o A372811 D = cosh( k*x*cosh(x*D +Ox));
%o A372811 T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
%o A372811 (2*n)! * polcoeff(C, 2*n, x)}
%o A372811 for(n=0, 30, print1( a(n), ", "))
%Y A372811 Cf. A370430 (k = 2), A372812 (S(x)), A372813 (D(x)), A372814 (T(x)), A143601.
%K A372811 nonn
%O A372811 0,3
%A A372811 _Paul D. Hanna_, May 16 2024