The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A014307 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A014307

Expansion of the e.g.f. sqrt(exp(x) / (2 - exp(x))).
(history; published version)

#144 by Michael De Vlieger at Wed Nov 15 08:03:19 EST 2023

STATUS

reviewed

approved

#143 by Stefano Spezia at Wed Nov 15 07:56:38 EST 2023

STATUS

proposed

reviewed

#142 by Seiichi Manyama at Wed Nov 15 07:52:58 EST 2023

STATUS

editing

proposed

#141 by Seiichi Manyama at Wed Nov 15 05:15:53 EST 2023

FORMULA

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (k/n - 2) * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 15 2023

STATUS

approved

editing

#140 by Michael De Vlieger at Sun Feb 27 16:12:27 EST 2022

STATUS

proposed

approved

#139 by Jon E. Schoenfield at Sun Feb 27 16:06:07 EST 2022

STATUS

editing

proposed

#138 by Jon E. Schoenfield at Sun Feb 27 16:05:37 EST 2022

FORMULA

E.g.f. B(x) = Integral_{t = 0..x} A(t) dt satisfies B'(x) = tan(B(x)) + sec(B(x)). (End)

Sum_{n>=1} a(n-1)*x^n/n! = series reversion (Integral_{t = 0..x} 1/(sec(t)+tan(t)) dt) = series reversion (Integral_{t = 0..x} (sec(t)-tan(t)) dt) = series reversion (x - x^2/2! + x^3/3! - 2*x^4/4! + 5*x^5/5! - 16*x^6/6! + ...) = x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 35*x^5/5! + 226*x^6/6! + ....

STATUS

reviewed

editing

Discussion

Sun Feb 27

16:06

Jon E. Schoenfield: Thanks. I'm sorry, I didn't catch these earlier.

#137 by Michel Marcus at Sun Feb 27 15:53:47 EST 2022

STATUS

proposed

reviewed

#136 by Jon E. Schoenfield at Sun Feb 27 15:27:59 EST 2022

STATUS

editing

proposed

Discussion

Sun Feb 27

15:53

Michel Marcus: yes

#135 by Jon E. Schoenfield at Sun Feb 27 15:27:52 EST 2022

FORMULA

G.f.: R(0)/(1-x), where R(k) = 1 - x^2*(k+1)*(2*k+1)/(x^2*(k+1)*(2*k+1) - (3*x*k+x-1)*(3*x*k+4*x-1)/R(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 30 2014

STATUS

proposed

editing