A380053 E.g.f. (exp(x) - 1)/cos(x).

1, 1, 4, 7, 36, 91, 624, 2087, 18256, 76231, 814144, 4078867, 51475776, 300870571, 4381112064, 29265244847, 482962852096, 3629392540111, 66942218896384, 558956224522027, 11394877025289216, 104659828714136851, 2336793875186479104, 23414201065072302407, 568240131312188379136

Offset: 1

Author

Paul D. Hanna, Jan 24 2025

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + 18256*x^9/9! + 76231*x^10/10! + ...
RELATED SERIES.
1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + 50521*x^10/10! + 2702765*x^12/12! + 199360981*x^14/14! + ... + A000364(n)*x^(2*n)/(2*n)! + ...
Let F(x) be the series reversion of e.g.f. A(x) so that A(F(x)) = x, then
F(x) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + ... + A380055(n)*x^n/n! + ...
where F(x) = log(1 + x*cos(F(x))).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A002084, A000364, A380054, A380055, A380057.

Programs

PARI

{a(n) = my(X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(X), n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.

(1) A(x) = (exp(x) - 1)/cos(x).

(2) A(F(x)) = x where F(x) = log(1 + x*cos(F(x))) equals the e.g.f. of A380055.

(3.a) a(2*n+1) = A002084(n) for n >= 0.

(3.b) a(2*n+1) = Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(n-k) for n >= 0.

(3.c) a(2*n) = Sum_{k=1..n} binomial(2*n, 2*k) * A000364(n-k) for n >= 1.

a(n) ~ (-1 + exp(Pi/2) + (-1 + exp(-Pi/2))*(-1)^n) * 2^(n + 3/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jan 24 2025