A370339 - OEIS

%I #22 Mar 02 2024 03:22:07

%S 1,1,29,2221,350489,93691801,38572439861,22632410010757,

%T 17988817024580273,18611994535647105841,24319398199016890981325,

%U 39169580455278059081821021,76249305758850982201026156233,176490984462982016166878127093961,479115505330137936930565874363849189

%N Expansion of e.g.f. A(x) satisfying A(x) = 1 / Product_{n>=1} cos(x^n*A(x)).

%C A related identity is sin(x)/x = Product_{n>=1} cos(x/2^n).

%C 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.

%C 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.

%C A related identity is (Pi/2) = 1 / (Sum_{n>=1} tan((Pi/2)/2^n) / 2^n).

%C lim_{n->oo} ( a(n)/(2*n)! )^(1/n) = 4.

%H Paul D. Hanna, <a href="/A370339/b370339.txt">Table of n, a(n) for n = 0..201</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! satisfies the following formulas.

%F (1) A(x) = 1 / Product_{n>=1} cos(x^n*A(x)).

%F (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.

%F (3) A(1/2) = Sum_{n>=0} a(n)/((2*n)!*4^n) = Pi/2.

%F a(n) ~ c * 16^n * (n-1)!^2, where c = 0.18032419917017939824645056... - _Vaclav Kotesovec_, Feb 29 2024

%e 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! + ...

%e where

%e A(x) = 1/(cos(x*A(x)) * cos(x^2*A(x)) * cos(x^3*A(x)) * cos(x^4*A(x)) * ...).

%e RELATED SERIES.

%e 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! + ...

%e where the logarithm of A(x) may be written as

%e 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!) + ...

%e in which the coefficients (A000182) are taken from the series for

%e 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)! + ...

%e SPECIFIC VALUES.

%e A(1/2) = Pi/2 = 1.570796326794896...

%e A(1/3) = 1.0767048225677796985699375012927448414632213680461...

%e A(1/4) = 1.0368864417986664841903383770323439256754564546659...

%e A(1/5) = 1.02215603157639512180398586667...

%e A(1/6) = 1.01489300419904897880469070874...

%e A(1/8) = 1.00811981394079053527600680239...

%e A(1/16) = 1.00197174861491784447864089136...

%e A(2/5) = 1.13484213008026434880482685706...

%e A(3/8) = 1.10876438339573451443889475275...

%e A(3/16) = 1.01922045277822853110544463913...

%e A(5/16) = 1.06434696931941878223868419402...

%e A(7/16) = 1.19092375254926233355508101258...

%e A(15/32) = 1.27051040050590151234289537354...

%e A(31/64) = 1.34043967392075568993521474964...

%e A(63/128) = 1.39800851071166904455009579306...

%e A(511/1024) = 1.5036464973982065250844881723...

%o (PARI) {a(n) = my(A=1); for(m=1,n+1, A=truncate(A);

%o A = 1 / prod(k=1,m, cos(x^k*A +O(x^(2*m+1))) ) ;); (2*n)!*polcoeff(A,2*n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A370436, A000182.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 29 2024