login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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!.
8

%I #25 Jan 17 2018 02:56:49

%S 0,1,-1,1,-2,4,-12,36,-144,576,-2880,14400,-86400,518400,-3628800,

%T 25401600,-203212800,1625702400,-14631321600,131681894400,

%U -1316818944000,13168189440000,-144850083840000,1593350922240000,-19120211066880000

%N 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!.

%C Signed version of A010551, with leading zero.

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

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

%F 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

%F 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

%F 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

%F Conjecture: 4*a(n) + 2*a(n-1) - (n-1)*(n-2)*a(n-2) = 0, n > 2. - _R. J. Mathar_, Nov 25 2015

%e 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.

%o (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]))}

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

%K sign

%O 0,5

%A _Paul D. Hanna_, Aug 23 2005