The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A111942

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A111942

Column 0 of the matrix logarithm (A111941) of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying the element in row n by n!.
(history; published version)

#25 by Bruno Berselli at Wed Jan 17 02:56:49 EST 2018

STATUS

reviewed

approved

#24 by Michel Marcus at Wed Jan 17 02:22:41 EST 2018

STATUS

proposed

reviewed

#23 by Jon E. Schoenfield at Wed Jan 17 01:29:14 EST 2018

STATUS

editing

proposed

#22 by Jon E. Schoenfield at Wed Jan 17 01:29:11 EST 2018

FORMULA

a(n) = (-1)^(n-1) * [floor((n-1)/2])! * [floor(n/2])! for n > 0, with a(0)=0.

E.g.f.: A(x) = (1-x/2)/sqrt(1-x^2/4)*acosarccos(1-x^2/2).

G.f.: x*G(0) where G(k) = 1 - (k+1)*x/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1) )) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 28 2012

G.f.: G(0)*x/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(1*k+1) - 1/(1 + 1/(1 - x*(k+1)/(x*(1*k+1) - 1/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013

G.f.: x/G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 07 2013

Conjecture: 4*a(n) + 2*a(n-1) - (n-1)*(n-2)*a(n-2) = 0, n > 2. - R. J. Mathar, Nov 25 2015

EXAMPLE

E.g.f.: A(x) = x - (1/2!)*x^2 + (1/3!)*x^3 - (2/4!)*x^4 + (4/5!)*x^5 - (12/6!)*x^6 + (36/7!)*x^7 - (144/8!)*x^8 + (576/9!)*x^9 + ... where A(x)*A(-x) = -acosarccos(1-x^2/2)^2.

STATUS

approved

editing

#21 by R. J. Mathar at Wed Nov 25 05:13:17 EST 2015

STATUS

proposed

approved

#20 by Michel Marcus at Wed Nov 25 04:39:52 EST 2015

STATUS

editing

proposed

#19 by Michel Marcus at Wed Nov 25 04:39:47 EST 2015

EXAMPLE

E.g.f.: A(x) = x - 1/2!*x^2 + 1/3!*x^3 - 2/4!*x^4 + 4/5!*x^5 - 12/6!*x^6 + 36/7!*x^7 - 144/8!*x^8 + 576/9!*x^9 + ... where A(x)*A(-x) = -acos(1-x^2/2)^2.

- 12/6!*x^6 + 36/7!*x^7 - 144/8!*x^8 + 576/9!*x^9 +...

where A(x)*A(-x) = -acos(1-x^2/2)^2.

#18 by Michel Marcus at Wed Nov 25 04:39:07 EST 2015

FORMULA

Conjecture: 4*a(n) +2*a(n-1) -(n-1)*(n-2)*a(n-2)=0, n>2 . - R. J. Mathar, Nov 25 2015

STATUS

proposed

editing

#17 by R. J. Mathar at Wed Nov 25 04:34:40 EST 2015

STATUS

editing

proposed

#16 by R. J. Mathar at Wed Nov 25 04:02:20 EST 2015

FORMULA

Conjecture: 4*a(n) +2*a(n-1) -(n-1)*(n-2)*a(n-2)=0, n>2 - R. J. Mathar, Nov 25 2015

STATUS

approved

editing