OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) = Sum_{j=0..n} A370432(n,j) * 2^(2*j).
E.g.f.: D(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.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r of e.g.f. D(x) is r = 0.458693345589772637742719473602361341151810356245785213... where D(r) = 2.216675597008249888019540624981069492182564304724769248...
EXAMPLE
E.g.f: 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! + ...
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
RELATED SERIES.
Related functions C(x), S(x), and T(x) are described below.
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! + ...
where C = cosh(x*D)
and C(x) = cosh( x*cosh(2*x*C(x)) ).
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! + ...
where S(x) = S = sinh(x*D)
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
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! + ...
where T(x) = (1/2) * sqrt(D^2 - 1)
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
D(1/3) = 1.276880244449228122993163054974488376796865611992370031...
D(1/4) = 1.138485942600540714616500323386982626365733417421170976...
D(1/5) = 1.085004369634098854421041251800873218914671999144038407...
D(1/6) = 1.057849764714936388260012199112395774792001649565003101...
D(1/10) = 1.020277074958546717842943931766605150247847706664020751...
PROG
(PARI) /* From D(x) = cosh( 2*x*cosh(x*D(x)) ) */
{a(n) = my(D=1); for(i=0, n, D=truncate(D); D = cosh( 2*x*cosh(x*D + x*O(x^(2*i))) ));
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
(PARI) /* From A370432 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2024
STATUS
approved