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.

A104981 Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.

Original entry on oeis.org

0, 1, 2, 7, 33, 191, 1297, 10063, 87669, 847015, 8989301, 103996703, 1303132269, 17589153719, 254509227541, 3931158238735, 64573130459613, 1124144767682215, 20677664894412965, 400760695386194687, 8163539437728923181
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Links

Crossrefs

Cf. A104980, A104982, A104986, A128175.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[nJean-François Alcover, Aug 09 2018 *)
  • PARI
    {a(n) = if(n<0, 0, (matrix(n+2, n+2, 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+1,2])}
  • 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,1) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

From Gary W. Adamson, Jul 14 2011: (Start)
Let M = triangle A128175 as an infinite square production matrix (deleting the first "1"):
1, 1, 0, 0, 0, ...
2, 2, 1, 0, 0, ...
4, 4, 3, 1, 0, ...
8, 8, 7, 4, 1, ...
...
a(n) = sum of top row terms of M^(n-1). Example: top row of M^4 = (71, 71, 38, 10, 1), sum = 191 = a(5). (End)
a(0) = 1, a(n) = n * a(n-1) + Sum_{j=1..n} A003319(j) * a(n - j), with offset 0 for the term 1. - F. Chapoton, Feb 26 2018

A104987 Row sums of triangle A104986, which equals the matrix logarithm of triangle A104980.

Original entry on oeis.org

0, 1, 4, 14, 58, 300, 1886, 13878, 116310, 1090500, 11296810, 128102714, 1578342010, 20998804576, 300081098918, 4584908039142, 74594230462318, 1287634918033836, 23506502407089874, 452508152936326482
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Links

Crossrefs

Cf. A104986, A104980.

Programs

  • Mathematica
    (* First program *)
nmax = 19;
M = Table[If[n==k, 0, If[n==k+1, -n+1, -Coefficient[(1 -1/Sum[i!*x^i, {i,0,n}])/x + O[x]^n, x, n-k-1]]], {n,1,nmax+1}, {k,1,nmax+1}];
T[n_, k_]/; 0<=k<=n:= Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p,n+1}]; T[, ] = 0;
a[n_]:= Sum[T[n, k], {k,0,n}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 09 2018, from PARI *)
(* Second program *)
t[n_, k_]:= t[n, k] = If[n=n, 0, t[n, k]], {n,0,q}, {k,0,q}]];
f[j_]:= f[j]= MatrixPower[M, j];
T[n_, k_]:= T[n, k]= If[k>n-1, 0, Sum[(-1)^(j-1)*f[j][[n+1, k+1]]/j, {j, n}]];
a[n_]:= a[n]= Sum[T[n, k], {k,0,n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jun 08 2021 *)
  • PARI
    {a(n)=sum(k=0,n,sum(p=1,n+1, (-1)^p*(matrix(n+1,n+1,m,j,if(m==j,0,if(m==j+1,-m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m),m-j-1))))^p)[n+1,k+1]/p))}

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.

A111560 Matrix logarithm of triangle A111553.

Original entry on oeis.org

0, 1, 0, 5, 2, 0, 34, 7, 3, 0, 282, 44, 10, 4, 0, 2696, 354, 60, 14, 5, 0, 28792, 3328, 470, 84, 19, 6, 0, 337072, 35144, 4344, 654, 118, 25, 7, 0, 4273632, 407984, 45320, 6008, 936, 164, 32, 8, 0, 58195072, 5137824, 521200, 62344, 8704, 1352, 224, 40, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Examples

			Triangle begins:
0;
1,0;
5,2,0;
34,7,3,0;
282,44,10,4,0;
2696,354,60,14,5,0;
28792,3328,470,84,19,6,0;
337072,35144,4344,654,118,25,7,0;
4273632,407984,45320,6008,936,164,32,8,0; ...

Crossrefs

Cf. A111553, A111557, A104986, A111541, A111549.

Programs

  • PARI
    {T(n,k)=local(M=matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))); sum(i=1,#M,(M^0-M)^i/i)[n+1,k+1]}
Showing 1-4 of 4 results.

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