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 1-7 of 7 results.

A144439 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 31, 31, 1, 1, 102, 326, 102, 1, 1, 317, 2406, 2406, 317, 1, 1, 964, 15087, 34336, 15087, 964, 1, 1, 2907, 86673, 380947, 380947, 86673, 2907, 1, 1, 8738, 473084, 3650206, 6925718, 3650206, 473084, 8738, 1, 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    31,      31,        1;
  1,   102,     326,      102,         1;
  1,   317,    2406,     2406,       317,         1;
  1,   964,   15087,    34336,     15087,       964,        1;
  1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
  1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
  1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144440, A144441, A144442, A144443, A144444, A144445, A144446, A144447.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n,  1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
Table[T[n,k,2,2], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 10 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144439(n,k): return T(n,k,2,2)
flatten([[A144439(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 10 2022

Formula

T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2.
Sum_{k=0..n} T(n, k) = s(n), where s(n) = 2*(n-1)*s(n-1) + 2*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = 4*3^(n-2) - (n+1).
T(n, 3) = (1/2)*(71*5^(n-3) - 8*(3*n+1)*3^(n-3) + n^2 + n - 1). (End)

A144440 Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 54, 54, 1, 1, 229, 789, 229, 1, 1, 932, 7975, 7975, 932, 1, 1, 3747, 68628, 161867, 68628, 3747, 1, 1, 15010, 543144, 2534759, 2534759, 543144, 15010, 1, 1, 60065, 4098439, 34243778, 66389335, 34243778, 4098439, 60065, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,     1;
  1,    11,       1;
  1,    54,      54,        1;
  1,   229,     789,      229,        1;
  1,   932,    7975,     7975,      932,        1;
  1,  3747,   68628,   161867,    68628,     3747,       1;
  1, 15010,  543144,  2534759,  2534759,   543144,   15010,     1;
  1, 60065, 4098439, 34243778, 66389335, 34243778, 4098439, 60065, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144441, A144442, A144443, A144444, A144445.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j] ];
Table[T[n,k,3,3], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144440(n,k): return T(n,k,3,3)
flatten([[A144440(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = (3*n-4)*s(n-1) + 3*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/3)*(11*4^(n-2) - (3*n+2)).
T(n, 3) = (1/18)*(9*n^2 + 3*n - 11 - 22*4^(n-3)*(12*n-1) + 709*7^(n-3)). (End)

A144441 Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 83, 83, 1, 1, 432, 1550, 432, 1, 1, 2181, 19898, 19898, 2181, 1, 1, 10930, 217887, 523548, 217887, 10930, 1, 1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1, 1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     14,        1;
  1,     83,       83,         1;
  1,    432,     1550,       432,         1;
  1,   2181,    19898,     19898,      2181,         1;
  1,  10930,   217887,    523548,    217887,     10930,        1;
  1,  54679,  2199237,  10589795,  10589795,   2199237,    54679,      1;
  1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144442, A144443, A144444, A144445.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
Table[T[n,k,4,4], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144441(n,k): return T(n,k,4,4)
flatten([[A144441(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(2*n-3)*s(n-1) + 4*s(n-2) with s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/2)*(7*5^(n-2) - (2*n+1)).
T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)

A144442 Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 118, 118, 1, 1, 729, 2681, 729, 1, 1, 4400, 41745, 41745, 4400, 1, 1, 26431, 555240, 1349245, 555240, 26431, 1, 1, 158622, 6816846, 33456685, 33456685, 6816846, 158622, 1, 1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     17,        1;
  1,    118,      118,         1;
  1,    729,     2681,       729,          1;
  1,   4400,    41745,     41745,       4400,         1;
  1,  26431,   555240,   1349245,     555240,     26431,        1;
  1, 158622,  6816846,  33456685,   33456685,   6816846,   158622,      1;
  1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144443, A144444, A144445.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
Table[T[n,k,5,5], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144442(n,k): return T(n,k,5,5)
flatten([[A144442(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = (5*n-8)*s(n-1) + 5*s(n-2), with s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/5)*(17*6^(n - 2) - (5*n + 2)).
T(n, 3) = (1/50)*(25*n^2 - 5*n - 31 - 34*6^(n - 3)*(30*n - 13) +
2489*11^(n - 3)). (End)

A144443 Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 159, 159, 1, 1, 1138, 4254, 1138, 1, 1, 7997, 77878, 77878, 7997, 1, 1, 56016, 1219167, 2984888, 1219167, 56016, 1, 1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 1, 1, 2745134, 244083268, 2219485106, 4400875078, 2219485106, 244083268, 2745134, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     20,        1;
  1,    159,      159,        1;
  1,   1138,     4254,     1138,        1;
  1,   7997,    77878,    77878,     7997,        1;
  1,  56016,  1219167,  2984888,  1219167,    56016,      1;
  1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144444, A144445.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
Table[T[n,k,6,6], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144443(n,k): return T(n,k,6,6)
flatten([[A144443(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Formula

T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(3*n-5)*s(n-1) + 6*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 04 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/3)*(10*7^(n-2) - (3*n+1)).
T(n, 3) = (1/18)*(9*n^2 -3*n -11 - 20*(21*n-11)*7^(n-3) + 997*13^(n-3)). (End)

A144445 Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 23, 1, 1, 206, 206, 1, 1, 1677, 6341, 1677, 1, 1, 13452, 133451, 133451, 13452, 1, 1, 107659, 2403612, 5916231, 2403612, 107659, 1, 1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1, 1, 6890633, 638151479, 5875203446, 11687580863, 5875203446, 638151479, 6890633, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     23,        1;
  1,    206,      206,         1;
  1,   1677,     6341,      1677,         1;
  1,  13452,   133451,    133451,     13452,        1;
  1, 107659,  2403612,   5916231,   2403612,   107659,      1;
  1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144444.

Programs

  • Mathematica
    T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j] ];
Table[T[n,k,7,7], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2022 *)
  • Sage
    def T(n,k,m,j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
def A144445(n,k): return T(n,k,7,7)
flatten([[A144445(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Formula

T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = (7*n-12)*s(n-1) + 7*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 04 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/7)*(23*8^(n-2) - (7*n+2)).
T(n, 3) = (1/98)*(49*n^2 - 21*n - 59 - 46*(56*n-33)*8^(n-3) + 5989*15^(n-3)). (End)

A144446 Array t(n, k) = (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k), with t(1, k) = 1, t(2, k) = 2, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 7, 2, 1, 30, 10, 2, 1, 157, 64, 13, 2, 1, 972, 532, 110, 16, 2, 1, 6961, 5448, 1249, 168, 19, 2, 1, 56660, 66440, 17816, 2416, 238, 22, 2, 1, 516901, 941056, 306619, 44160, 4141, 320, 25, 2, 1, 5225670, 15189776, 6185828, 981184, 92292, 6532, 414, 28, 2, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Keywords

Examples

			Array t(n,k) begins as:
    1,    1,     1,     1,     1,      1, ...;
    2,    2,     2,     2,     2,      2, ...;
    7,   10,    13,    16,    19,     22, ...;
   30,   64,   110,   168,   238,    320, ...;
  157,  532,  1249,  2416,  4141,   6532, ...;
  972, 5448, 17816, 44160, 92292, 171752, ...;
Antidiagonal triangle T(n,k) begins as:
        1;
        2,        1;
        7,        2,       1;
       30,       10,       2,      1;
      157,       64,      13,      2,     1;
      972,      532,     110,     16,     2,    1;
     6961,     5448,    1249,    168,    19,    2,   1;
    56660,    66440,   17816,   2416,   238,   22,   2,  1;
   516901,   941056,  306619,  44160,  4141,  320,  25,  2, 1;
  5225670, 15189776, 6185828, 981184, 92292, 6532, 414, 28, 2, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144444, A144445, A144447.
Cf. A001053.

Programs

  • Magma
    function T(n,k) // triangle form; A144446
  if k gt n-2 then return n-k+1;
  else return (k*(n-k)+2-k)*T(n-1, k) + k*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2022
  • Mathematica
    t[n_, k_]:= t[n, k]= If[n<3, n, (k*(n-1) +2-k)*t[n-1,k] + k*t[n-2,k]];
T[n_, k_]:= t[n-k+1,k];
Table[T[n, k], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2022 *)
  • Sage
    def t(n,k): return n if(n<3) else (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k)
def A144446(n,k): return t(n-k+1,k)
flatten([[A144446(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 05 2022

Formula

T(n, k) = t(n-k+1, k), where t(n, k) = (k*(n-1) +2-k)*t(n-1, k) + k*t(n-2, k) with t(1, k) = 1, t(2, k) = 2.
T(n, 1) = A001053(n+1).
T(n, k) = (k*(n-k)+2-k)*T(n-1, k) + k*T(n-2, k) with T(n, n-1) = 2, T(n, n) = 1 (as a triangle). - G. C. Greubel, Mar 05 2022

Extensions

Edited by G. C. Greubel, Mar 05 2022
Showing 1-7 of 7 results.

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