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 1-4 of 4 results.

A104989 Row sums of triangle A104988, which equals the matrix square of triangle A104980.

Original entry on oeis.org

1, 3, 13, 69, 433, 3133, 25657, 234537, 2367825, 26176981, 314670353, 4088360569, 57112939433, 853922061413, 13609089281849, 230346936181465, 4127180489763649, 78046835384582069, 1553536327234953153
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Links

Crossrefs

Cf. A104980, A104988.

Programs

  • Mathematica
    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, 2];
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 *)
  • PARI
    {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))))^-2)[n+1,k+1]))}

Formula

a(n) = Sum_{k=0..n} A104988(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.

A104982 Column 3 of triangle A104980, omitting leading zeros.

Original entry on oeis.org

1, 4, 21, 133, 977, 8135, 75609, 775667, 8707057, 106185715, 1398451353, 19786121467, 299384925569, 4825081148819, 82531968286569, 1493412479919371, 28504390805515921, 572363196501249667, 12061937537478658809
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Equals one-half of column 0 (after initial term) in triangle A104988, which equals the matrix square of triangle A104980.

Links

Crossrefs

Cf. A104980, A104981, A104988.

Programs

  • Mathematica
    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}]]]];
Table[T[n+3, 3], {n, 0, 30}] (* G. C. Greubel, Jun 07 2021 *)
  • PARI
    {a(n) = if(n<0, 0, (matrix(n+4, n+4, 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))))^-1)[n+4,4])}
  • 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) )
[T(n+3,3) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

a(n) = A104988(n+1, 0)/2 for n>=0.

A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 20, 5, 1, 3447, 1297, 461, 120, 28, 6, 1, 29093, 10063, 3447, 836, 181, 38, 7, 1, 273343, 87669, 29093, 6616, 1333, 270, 49, 8, 1, 2829325, 847015, 273343, 58576, 11029, 2150, 375, 61, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 17 2009

Keywords

Examples

			To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums.
This square array A begins:
(1), (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, 1, 1, 1, 1, ...;
(1), (2), 3, 4, (5), 6, 7, 8, (9), 10, 11, 12, 13, (14), 15, 16, ...;
(3), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...;
(13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...;
(71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...;
(461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...;
(3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...;
(29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...;
(273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...;
where terms in parenthesis at positions {0,1,4,8,13,..} in a row
are removed before taking partial sums to obtain the next row.
...
RELATION TO SPECIAL TRIANGLE.
Triangle A104980 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; ...
in which column 0 and column 1 are found in square array A.
...
Matrix square of A104980 = triangle A104988 which begins:
1;
2, 1;
8, 4, 1;
42, 20, 6, 1;
266, 120, 38, 8, 1;
1954, 836, 270, 62, 10, 1;
16270, 6616, 2150, 516, 92, 12, 1;
151218, 58576, 19030, 4688, 882, 128, 14, 1; ...
where column 1 and column 2 are also found in square array A.

Crossrefs

Cf. columns: A003319, A104981, A156629, related triangles: A104980, A104988.
Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217.

Programs

  • PARI
    {T (n, k)=local (A=0, b=2, c=1, d=0); if (n==0, A=1, until (d>k, if (c==b* (b+1)/2-2, b+=1, A+=T (n-1, c); d+=1); c+=1)); A}

Formula

Column 0 = Column 0 of triangle A104980 = A003319.
Column 1 = Column 1 of triangle A104980 = A104981.
Column 3 = column 1 of A104988 (matrix square of A104980).
Column 5 = column 2 of A104988 (matrix square of A104980).
Showing 1-4 of 4 results.

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