cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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!.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Comments

Signed version of A010551, with leading zero.

Examples

			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.

Crossrefs

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

Programs

  • 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]))}

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

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 1, 1, 0, -2, -1, -1, -1, 0, 4, 2, 1, 1, 1, 0, -12, -4, -2, -1, -1, -1, 0, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -86400, -14400, -2880, -576, -144, -36
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Examples

			Triangle begins:
0;
1, 0;
-1, -1, 0;
1, 1, 1, 0;
-2, -1, -1, -1, 0;
4, 2, 1, 1, 1, 0;
-12, -4, -2, -1, -1, -1, 0;
36, 12, 4, 2, 1, 1, 1, 0;
-144, -36, -12, -4, -2, -1, -1, -1, 0;
576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
where, apart from signs, the columns are all the same (A111942).
...
Triangle A111940 begins:
1;
1, 1;
-1, -1, 1;
0, 0, 1, 1;
0, 0, -1, -1, 1;
0, 0, 0, 0, 1, 1;
0, 0, 0, 0, -1, -1, 1;
0, 0, 0, 0, 0, 0, 1 ,1;
0, 0, 0, 0, 0, 0, -1, -1, 1; ...
where the matrix inverse shifts columns left and up one place.
...
The matrix log of A111940, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, -1/1!, 0;
1/3!, 1/2!, 1/1!, 0;
-2/4!, -1/3!, -1/2!, -1/1!, 0;
4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
Note that the square of the matrix log of A111940 begins:
0;
0, 0;
-1, 0, 0;
0, -1, 0, 0;
-1/12, 0, -1, 0, 0;
0, -1/12, 0, -1, 0, 0;
-1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
where nonzero terms are negative unit fractions with denominators given by A002544:
[1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...].

Crossrefs

Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).

Programs

  • PARI
    {T(n,k,q=-1) = local(A=Mat(1),B); if(n

Formula

T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.

A110503 Triangle, read by rows, which shifts one column left under matrix inverse.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -2, 1, 1, 1, -1, 1, -1, 1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2005

Keywords

Comments

The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.

Examples

			Triangle begins:
  1;
  1,  1;
  1, -1,  1;
  1, -2,  1,  1;
  1, -1,  1, -1,  1;
  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -2,  1,  1;
  1, -1,  1, -1,  1, -1,  1, -1,  1;
  1, -1,  1, -1,  1, -1,  1, -2,  1,  1; ...
The matrix inverse drops the first column:
   1;
  -1,  1;
  -2,  1,  1;
  -1,  1, -1,  1;
  -1,  1, -2,  1,  1;
  -1,  1, -1,  1, -1,  1; ...
The matrix logarithm equals:
     0;
    1/1!,     0;
    3/2!,   -1/1!,   0;
    7/3!,   -3/2!,  1/1!,   0;
   30/4!,   -7/3!,  3/2!, -1/1!,  0;
  144/5!,  -30/4!,  7/3!, -3/2!, 1/1!,   0;
  876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...
unsigned columns of which all equal A110505.

Crossrefs

Cf. A110504 (matrix log), A110505 (column 0 of log).
Cf. A111940 (variant).

Programs

  • PARI
    T(n,k)=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))[n+1,k+1]

Formula

T(n, k) = +1 when k == 0 (mod 2), T(n, k)=-1 when k == 1 (mod 2), except for T(k+2, k) = -2 when k == 1 (mod 2) and T(n, n) = 1.
G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*arccos(1-x^2/2)) + (-1)^k*sin(m*arccos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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