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!.

0, 1, -1, 1, -2, 4, -12, 36, -144, 576, -2880, 14400, -86400, 518400, -3628800, 25401600, -203212800, 1625702400, -14631321600, 131681894400, -1316818944000, 13168189440000, -144850083840000, 1593350922240000, -19120211066880000

Offset

0,5

Comments

Signed version of A010551, with leading zero.

Links

Table of n, a(n) for n=0..24.

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)*arccos(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) = -arccos(1-x^2/2)^2.

Prog

(PARI) {a(n, q=-1)=local(A=Mat(1), B); if(n<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, if(j==1, B[i, j]=(A^q)[i-1, 1], B[i, j]=(A^q)[i-1, j-1])); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return(n!*B[n+1, 1]))}

Crossrefs

Cf. A111940 (triangle), A111941 (matrix log), A110505 (variant), A010551 (unsigned).

Sequence in context: A195733 A046993 A282165 * A010551 A349282 A276230

Adjacent sequences: A111939 A111940 A111941 * A111943 A111944 A111945

Keyword

sign

Author

Paul D. Hanna, Aug 23 2005

Status

approved