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.

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 22, 8, 3, 1, 148, 44, 14, 4, 1, 1156, 296, 84, 22, 5, 1, 10192, 2312, 600, 148, 32, 6, 1, 99688, 20384, 4908, 1156, 242, 44, 7, 1, 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1, 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

Column 0 equals A111529 (related to log of factorial series).
Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.

Examples

			SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);
SHIFT_LEFT(column 0 of log(T)) = column 2 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1, 1;
4, 2, 1;
22, 8, 3, 1;
148, 44, 14, 4, 1;
1156, 296, 84, 22, 5, 1;
10192, 2312, 600, 148, 32, 6, 1;
99688, 20384, 4908, 1156, 242, 44, 7, 1;
1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1;
12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1; ...
...
After initial term, column 1 is twice column 0.
Matrix inverse T^-1 = A111540 starts:
1;
-1, 1;
-2, -2, 1;
-8, -2, -3, 1;
-44, -8, -2, -4, 1;
-296, -44, -8, -2, -5, 1;
-2312, -296, -44, -8, -2, -6, 1;
-20384, -2312, -296, -44, -8, -2, -7, 1;
-199376, -20384, -2312, -296, -44, -8, -2, -8, 1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 1 of T.
Matrix logarithm log(T) = A111541 is:
0;
1, 0;
3, 2, 0;
14, 5, 3, 0;
84, 22, 8, 4, 0;
600, 128, 36, 12, 5, 0;
4908, 896, 212, 58, 17, 6, 0;
44952, 7220, 1496, 360, 90, 23, 7, 0;
454344, 65336, 12128, 2652, 602, 134, 30, 8, 0;
5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0; ...
compare column 0 of log(T) to column 2 of T.

Links

Crossrefs

Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = If[nJean-François Alcover, Jan 24 2017, adapted from PARI *)
  • PARI
    T(n,k)=if(n

Formula

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.

A111548 Matrix inverse of triangle A111544.

Original entry on oeis.org

1, -1, 1, -3, -2, 1, -15, -3, -3, 1, -99, -15, -3, -4, 1, -783, -99, -15, -3, -5, 1, -7083, -783, -99, -15, -3, -6, 1, -71415, -7083, -783, -99, -15, -3, -7, 1, -789939, -71415, -7083, -783, -99, -15, -3, -8, 1, -9485343, -789939, -71415, -7083, -783, -99, -15, -3, -9, 1, -122721723, -9485343, -789939, -71415
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

The column sequences are all equal after the initial terms and are derived from the logarithm of a factorial series (cf. A111546).

Examples

			Triangle begins:
1;
-1,1;
-3,-2,1;
-15,-3,-3,1;
-99,-15,-3,-4,1;
-783,-99,-15,-3,-5,1;
-7083,-783,-99,-15,-3,-6,1;
-71415,-7083,-783,-99,-15,-3,-7,1;
-789939,-71415,-7083,-783,-99,-15,-3,-8,1; ...

Crossrefs

Cf. A111544, A111546, A111540.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n, n)=1 and T(n+1, n)=-n-1, else T(n+k+1, k) = -A111546(k) for k>=1.

A111559 Matrix inverse of triangle A111553.

Original entry on oeis.org

1, -1, 1, -4, -2, 1, -24, -4, -3, 1, -184, -24, -4, -4, 1, -1664, -184, -24, -4, -5, 1, -17024, -1664, -184, -24, -4, -6, 1, -192384, -17024, -1664, -184, -24, -4, -7, 1, -2366144, -192384, -17024, -1664, -184, -24, -4, -8, 1, -31362304, -2366144, -192384, -17024, -1664, -184, -24, -4, -9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

After initial terms, all columns are equal to -A111556.

Examples

			Triangle begins:
1;
-1,1;
-4,-2,1;
-24,-4,-3,1;
-184,-24,-4,-4,1;
-1664,-184,-24,-4,-5,1;
-17024,-1664,-184,-24,-4,-6,1;
-192384,-17024,-1664,-184,-24,-4,-7,1; ...

Crossrefs

Cf. A111553, A111556, A104984, A111540, A111548.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n, n)=1 and T(n+1, n)=-n-1, else T(n+k+1, k) = -A111556(k) for k>=1.
Showing 1-3 of 3 results.

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