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.

A104985 Row sums of triangle A104984.

Original entry on oeis.org

1, 0, -2, -6, -20, -92, -554, -4002, -33096, -306440, -3135766, -35134670, -427878628, -5628940084, -79572364498, -1203168642362, -19379896959776, -331331041788640, -5993029816637262, -114348894263852326, -2295445815821635932, -48362099044178487564
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

A104984 equals the matrix inverse of A104980.

Links

Crossrefs

Cf. A104984, A104980.

Programs

  • Mathematica
    A003319[n_]:= A003319[n]= If[n==0, 0, n! -Sum[j!*A003319[n-j], {j,n-1}]];
A104984[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];
a[n_]:= Sum[A104984[n, k], {k,0,n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 07 2021 *)
  • PARI
    {a(n)=sum(k=0,n,if(k==n,1,if(k==n-1,-n, -polcoeff((1-1/sum(i=0,n-k,i!*x^i))/x+O(x^(n-k)),n-k-1) )))}
  • Sage
    @CachedFunction
def T(n,k):
    if (k==n): return 1
    elif (k==n-1): return -n
    else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

a(n) = Sum_{k=0..n} A104984(n, k).

A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 21, 5, 1, 3447, 1297, 461, 133, 31, 6, 1, 29093, 10063, 3447, 977, 225, 43, 7, 1, 273343, 87669, 29093, 8135, 1859, 353, 57, 8, 1, 2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Column 0 equals A003319 (indecomposable permutations). Amazingly, column 1 (A104981) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A104986.
From Paul D. Hanna, Feb 17 2009: (Start)
Square array A156628 has columns found in this triangle T:
Column 0 of A156628 = column 0 of T = A003319;
Column 1 of A156628 = column 1 of T = A104981;
Column 2 of A156628 = column 2 of T = A003319 shifted;
Column 3 of A156628 = column 1 of T^2 (A104988);
Column 5 of A156628 = column 2 of T^2 (A104988). (End)

Examples

			SHIFT_LEFT(column 0 of T^-1) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^1) = 1*(column 2 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 3 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
        1;
        1,      1;
        3,      2,      1;
       13,      7,      3,     1;
       71,     33,     13,     4,     1;
      461,    191,     71,    21,     5,    1;
     3447,   1297,    461,   133,    31,    6,   1;
    29093,  10063,   3447,   977,   225,   43,   7,  1;
   273343,  87669,  29093,  8135,  1859,  353,  57,  8, 1;
  2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1; ...
Matrix inverse T^-1 is A104984 which begins:
     1;
    -1,   1;
    -1,  -2,   1;
    -3,  -1,  -3,  1;
   -13,  -3,  -1, -4,  1;
   -71, -13,  -3, -1, -5,  1;
  -461, -71, -13, -3, -1, -6, 1; ...
Matrix T also satisfies:
[I + SHIFT_LEFT(T)] = [I - SHIFT_DOWN(T)]^-1, which starts:
    1;
    1,  1;
    2,  1,  1;
    7,  3,  1, 1;
   33, 13,  4, 1, 1;
  191, 71, 21, 5, 1, 1; ...
where SHIFT_DOWN(T) shifts columns of T down 1 row,
and SHIFT_LEFT(T) shifts rows of T left 1 column,
with both operations leaving zeros in the diagonal.

Links

Crossrefs

Cf. A003319 (column 0), A104981 (column 1), A104983 (row sums), A104984 (matrix inverse), A104988 (matrix square), A104990 (matrix cube), A104986 (matrix log), A156628.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[nJean-François Alcover, Aug 09 2018, from PARI *)
  • PARI
    {T(n,k) = if(n
  • PARI
    {T(n,k) = if(n
  • Sage
    @CachedFunction
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==n-1): return n
    else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 07 2021

Formula

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

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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