OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..150
FORMULA
E.g.f.: Series_Reversion( Integral cos(x)/cosh(x) dx ).
EXAMPLE
E.g.f.: A(x) = x + 2*x^3/3! + 28*x^5/5! + 1048*x^7/7! + 75792*x^9/9! + 8997152*x^11/11! + 1589002688*x^13/13! + 390961266048*x^15/15! + 127846741426432*x^17/17! + 53632884600381952*x^19/19! +...
such that A'(x) = cosh(A(x)) / cos(A(x)).
RELATED SERIES.
Let B(x) be the series reversion of e.g.f. A(x), then
B(x) = x - 2*x^3/3! + 12*x^5/5! - 152*x^7/7! + 3472*x^9/9! - 126752*x^11/11! + 6781632*x^13/13! - 500231552*x^15/15! +...+ A000795(n-1)*x^(2*n-1)!/(2*n-1)! +...
then G'(x) = cos(x)/cosh(x).
Let G(x) be defined by G(G(x)) = A(x), then
G(x) = x + x^3/3! + 9*x^5/5! + 237*x^7/7! + 12385*x^9/9! + 1067225*x^11/11! + 136228105*x^13/13! + 24056468229*x^15/15! + 5614204466945*x^17/17! + 1677288189454257*x^19/19! + 626137638928559689*x^21/21! + 285873599602408829469*x^23/23! + 156375718123032150293473*x^25/25! +...
PROG
(PARI) {a(n) = my(A=x, Ox=x*O(x^(2*n))); for(i=0, n, A = intformal( cosh(A +Ox) / cos(A +Ox))); (2*n-1)!*polcoeff( G = A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 23 2017
STATUS
approved