A104990 -id:A104990 - cp's OEIS frontend

A104991 Row sums of triangle A104990, which equals the matrix cube of triangle A104980.

1, 4, 22, 142, 1048, 8704, 80386, 817786, 9093340, 109794556, 1431360958, 20047830262, 300343272952, 4793871035416, 81232799446906, 1456671526257106, 27562347560513524, 548844246683051860, 11474015910364016086

Offset: 0

Paul D. Hanna, Apr 10 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A104980, A104990.

nmax:=30;
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*T[n, k+1] + Sum[T[j, 0]*T[n, j+k+1], {j, 0, n-k-1}]]]]; (* T=A104980 *)
M:= M= With[{q = nmax}, Table[If[j>i, 0, T[i,j]], {i,0,q}, {j,0,q}]];
f:= f= MatrixPower[M, 3];
a[n_]:= a[n]= Sum[f[[n+1, k+1]], {k,0,n}];
Table[a[n], {n, 0, nmax}] (* G. C. Greubel, Jun 08 2021 *)

{a(n)=if(n<0,0,sum(k=0,n,(matrix(n+1,n+1,m,j,if(m==j,1,if(m==j+1,-m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m),m-j-1))))^-3)[n+1,k+1]))}

a(n) = Sum_{k=0..n} A104990(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) = pT(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

Paul D. Hanna, Apr 10 2005

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)

			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.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

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

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
				
```

@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

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.

cp's OEIS Frontend

A104991 Row sums of triangle A104990, which equals the matrix cube of triangle A104980.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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) = pT(p+m,p+1) for all m>=1 and p>=-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

cp's OEIS Frontend

A104991 Row sums of triangle A104990, which equals the matrix cube of triangle A104980.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

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) = pT(p+m,p+1) for all m>=1 and p>=-1.