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 1-10 of 15 results. Next

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

A104986 Matrix logarithm of triangle A104980.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 7, 4, 3, 0, 33, 14, 7, 4, 0, 191, 66, 27, 11, 5, 0, 1297, 382, 137, 48, 16, 6, 0, 10063, 2594, 843, 270, 79, 22, 7, 0, 87669, 20126, 6041, 1820, 495, 122, 29, 8, 0, 847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0, 8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Column 0 equals column 1 of triangular matrix A104980, which satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0. Column 1 equals twice column 0.

Examples

			Triangle begins:
        0;
        1,       0;
        2,       2,      0;
        7,       4,      3,      0;
       33,      14,      7,      4,     0;
      191,      66,     27,     11,     5,    0;
     1297,     382,    137,     48,    16,    6,    0;
    10063,    2594,    843,    270,    79,   22,    7,   0;
    87669,   20126,   6041,   1820,   495,  122,   29,   8,  0;
   847015,  175338,  49219,  14176,  3679,  848,  179,  37,  9,  0;
  8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0; ...

Links

Crossrefs

Cf. A104980, A104981 (column 0), A104987 (row sums).

Programs

  • Mathematica
    nmax = 10;
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, 1, n+1}]; T[, ] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)
  • PARI
    T(n,k)=if(n

Formula

T(n, 0) = A104981(n), T(n+1, 1) = 2*T(n, 0) for n>=0.

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))}

A104988 Matrix square of triangle A104980.

Original entry on oeis.org

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, 1551334, 573672, 185674, 46516, 9050, 1392, 170, 16, 1, 17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

Examples

			Triangle 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;
   1551334,  573672,  185674,  46516,  9050,  1392,  170,  16,  1;
  17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 1;

Links

Crossrefs

Cf. A104980, A104982 (column 0), A104989 (row sums).

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}]]]]; (* t = A104980 *)
M:= With[{q=20}, Table[If[j>i, 0, t[i, j]], {i,0,q}, {j,0,q}]];
Table[MatrixPower[M, 2][[n+1, k+1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)
  • PARI
    T(n,k)= if(n

Formula

T(n+1, 0) = 2*A104980(n+3, 3) = 2*A104982(n) 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.

A104984 Matrix inverse of triangle A104980.

Original entry on oeis.org

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, -3447, -461, -71, -13, -3, -1, -7, 1, -29093, -3447, -461, -71, -13, -3, -1, -8, 1, -273343, -29093, -3447, -461, -71, -13, -3, -1, -9, 1, -2829325, -273343, -29093, -3447, -461, -71, -13, -3, -1, -10, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Inverse matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

Examples

			Triangle 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;
   -3447,  -461,  -71, -13,  -3, -1, -7,  1;
  -29093, -3447, -461, -71, -13, -3, -1, -8, 1; ...

Links

Crossrefs

Cf. A104980, A104985 (row sums).

Programs

  • Mathematica
    A003319[n_]:= A003319[n]= If[n==0, 0, n! - Sum[j!*A003319[n-j], {j,n-1}]];
T[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)
  • PARI
    T(n,k)=if(n==k,1,if(n==k+1,-n,-(n-k)!-sum(i=1,n-k-1,i!*T(n-k-i,0))));
  • 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) )
[[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2021

Formula

T(n, n) = 1, T(n+1, n) = -(n+1) for n >= 0; otherwise T(n, k) = T(n-k, 0) = -A003319(n-k-1) for n > k+1 and k >= 0.
Sum_{k=0..n} T(n, k) = A104985(n). - G. C. Greubel, Jun 07 2021

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).

A104990 Matrix cube of triangle A104980.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 93, 39, 9, 1, 675, 285, 75, 12, 1, 5577, 2331, 657, 123, 15, 1, 51555, 21153, 6207, 1269, 183, 18, 1, 526809, 211227, 63549, 13743, 2181, 255, 21, 1, 5895819, 2304321, 704319, 158325, 26739, 3453, 339, 24, 1, 71733585, 27291843, 8424813, 1947711, 343641, 47355, 5145, 435, 27, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Comments

Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

Examples

			Triangle begins:
         1;
         3,        1;
        15,        6,       1;
        93,       39,       9,       1;
       675,      285,      75,      12,      1;
      5577,     2331,     657,     123,     15,     1;
     51555,    21153,    6207,    1269,    183,    18,    1;
    526809,   211227,   63549,   13743,   2181,   255,   21,   1;
   5895819,  2304321,  704319,  158325,  26739,  3453,  339,  24,  1;
  71733585, 27291843, 8424813, 1947711, 343641, 47355, 5145, 435, 27, 1;

Links

Crossrefs

Cf. A104980, A104982 (column 0), A104991 (row sums).

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}]]]]; (* t = A104980 *)
M:= With[{q=20}, Table[If[j>i, 0, t[i, j]], {i,0,q}, {j,0,q}]];
Table[MatrixPower[M, 3][[n+1, k+1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)
  • PARI
    T(n,k)=if(n

Formula

T(n+1, 0) = 3*A104980(n+4, 4) for n>=0.

A104983 Row sums of triangular matrix T = A104980 which satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T).

Original entry on oeis.org

1, 2, 6, 24, 122, 750, 5376, 43856, 400518, 4046334, 44808104, 539850984, 7032370302, 98516491214, 1477264979352, 23612920280976, 400847064718166, 7202901369491694, 136596819590256984, 2726503675380494408
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Links

Crossrefs

Cf. A104980.

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}]]]]; (* T=A104980 *)
Table[Sum[T[n, k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 07 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))))^-1)[n+1,k+1]))};
  • 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) )
[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} A104980(n, k).

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

Original entry on oeis.org

1, 4, 22, 142, 1048, 8704, 80386, 817786, 9093340, 109794556, 1431360958, 20047830262, 300343272952, 4793871035416, 81232799446906, 1456671526257106, 27562347560513524, 548844246683051860, 11474015910364016086
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2005

Keywords

Links

Crossrefs

Cf. A104980, A104990.

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, 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 *)
  • 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))))^-3)[n+1,k+1]))}

Formula

a(n) = Sum_{k=0..n} A104990(n, k).
Showing 1-10 of 15 results. Next

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