OFFSET
0,3
COMMENTS
A related identity is sin(x)/x = Product_{n>=1} cos(x/2^n).
Motivated by the fixed point (Pi/2) = 1 / Product_{n>=1} cos((Pi/2)*(1/2)^n), from a formula added by Ilya Gutkovskiy to A000796 on Aug 07 2016.
The radius of convergence of e.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! is 1/2, and A(x) at x = 1/2 converges to A(1/2) = Pi/2.
A related identity is (Pi/2) = 1 / (Sum_{n>=1} tan((Pi/2)/2^n) / 2^n).
lim_{n->oo} ( a(n)/(2*n)! )^(1/n) = 4.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..201
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = 1 / Product_{n>=1} cos(x^n*A(x)).
(2) log(A(x)) = Sum_{n>=1} A000182(n) * A(x)^(2*n) * x^(2*n)/((1-x^(2*n))*(2*n)!), where A000182 are the tangent numbers.
(3) A(1/2) = Sum_{n>=0} a(n)/((2*n)!*4^n) = Pi/2.
a(n) ~ c * 16^n * (n-1)!^2, where c = 0.18032419917017939824645056... - Vaclav Kotesovec, Feb 29 2024
EXAMPLE
G.f.: A(x) = 1 + x^2/2! + 29*x^4/4! + 2221*x^6/6! + 350489*x^8/8! + 93691801*x^10/10! + 38572439861*x^12/12! + 22632410010757*x^14/14! + ...
where
A(x) = 1/(cos(x*A(x)) * cos(x^2*A(x)) * cos(x^3*A(x)) * cos(x^4*A(x)) * ...).
RELATED SERIES.
log(A(x)) = x^2/2! + 26*x^4/4! + 1816*x^6/6! + 270416*x^8/8! + 69316096*x^10/10! + 27774564608*x^12/12! + 15980811489280*x^14/14! + ...
where the logarithm of A(x) may be written as
log(A(x)) = A(x)^2*x^2/((1-x^2)*2!) + 2*A(x)^4*x^4/((1-x^4)*4!) + 16*A(x)^6*x^6/((1-x^6)*6!) + 272*A(x)^8*x^8/((1-x^8)*8!) + ...
in which the coefficients (A000182) are taken from the series for
log(1/cos(x)) = x^2/2! + 2*x^4/4! + 16*x^6/6! + 272*x^8/8! + 7936*x^10/10! + 353792*x^12/12! + ... + A000182(n)*x^(2*n)/(2*n)! + ...
SPECIFIC VALUES.
A(1/2) = Pi/2 = 1.570796326794896...
A(1/3) = 1.0767048225677796985699375012927448414632213680461...
A(1/4) = 1.0368864417986664841903383770323439256754564546659...
A(1/5) = 1.02215603157639512180398586667...
A(1/6) = 1.01489300419904897880469070874...
A(1/8) = 1.00811981394079053527600680239...
A(1/16) = 1.00197174861491784447864089136...
A(2/5) = 1.13484213008026434880482685706...
A(3/8) = 1.10876438339573451443889475275...
A(3/16) = 1.01922045277822853110544463913...
A(5/16) = 1.06434696931941878223868419402...
A(7/16) = 1.19092375254926233355508101258...
A(15/32) = 1.27051040050590151234289537354...
A(31/64) = 1.34043967392075568993521474964...
A(63/128) = 1.39800851071166904455009579306...
A(511/1024) = 1.5036464973982065250844881723...
PROG
(PARI) {a(n) = my(A=1); for(m=1, n+1, A=truncate(A);
A = 1 / prod(k=1, m, cos(x^k*A +O(x^(2*m+1))) ) ; ); (2*n)!*polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 29 2024
STATUS
approved