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.

Previous Showing 11-20 of 20 results.

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)

A144444 Triangle read by rows: T(n, k) = (1-n+k)*T(n-1, k-1) + (2-k)*T(n-1, k) - T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -2, -2, 1, 1, -3, 5, -3, 1, 1, -4, 3, 3, -4, 1, 1, -5, 12, -17, 12, -5, 1, 1, -6, 12, -5, -5, 12, -6, 1, 1, -7, 23, -50, 47, -50, 23, -7, 1, 1, -8, 25, -27, 64, 64, -27, 25, -8, 1
Offset: 1

Views

Author

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

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1, -1,  1;
  1, -2, -2,   1;
  1, -3,  5,  -3,  1;
  1, -4,  3,   3, -4,   1;
  1, -5, 12, -17, 12,  -5,   1;
  1, -6, 12,  -5, -5,  12,  -6,  1;
  1, -7, 23, -50, 47, -50,  23, -7,  1;
  1, -8, 25, -27, 64,  64, -27, 25, -8, 1;

Links

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144445.
Cf. A075374.

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,-1,-1], {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 A144444(n,k): return T(n,k,-1,-1)
flatten([[A144444(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Formula

T(n, k) = (1-n+k)*T(n-1, k-1) + (2-k)*T(n-1, k) - 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) = -(n-4)*s(n-1) - s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 04 2022: (Start)
Sum_{k=1..n} T(n, k) = 2*[n<3] + (-1)^(n-1)*A075374(n-2).
T(n, n-k) = T(n, k).
T(n, 2) = [n=2] - n + 2.
T(n, 3) = (1/2)*((n^2 -5*n +5) -5*(-1)^n) - [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)

A142461 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 111, 111, 1, 1, 796, 2886, 796, 1, 1, 5597, 52642, 52642, 5597, 1, 1, 39210, 824271, 2000396, 824271, 39210, 1, 1, 274507, 11931033, 58614299, 58614299, 11931033, 274507, 1, 1, 1921592, 165260188, 1483533704, 2930714950, 1483533704, 165260188, 1921592, 1
Offset: 1

Views

Author

Roger L. Bagula, Sep 19 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     14,        1;
  1,    111,      111,        1;
  1,    796,     2886,      796,        1;
  1,   5597,    52642,    52642,     5597,        1;
  1,  39210,   824271,  2000396,   824271,    39210,      1;
  1, 274507, 11931033, 58614299, 58614299, 11931033, 274507, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, ...
Cf. A047657 (row sums).

Programs

  • Mathematica
    T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m]];
A142461[n_, k_]:= T[n, k, 6];
Table[A142461[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A142461(n,k): return T(n,k,6)
flatten([[ A142461(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n,k,m) = (m*n - m*k + 1)*T(n-1, k-1, m) + (m*k - (m-1))*T(n-1, k, m), with T(n, 1, m) = T(n, n, m) = 1, and m = 6.
Sum_{k=1..n} T(n, k, 6) = A047657(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013

A142462 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.

Original entry on oeis.org

1, 1, 1, 1, 16, 1, 1, 143, 143, 1, 1, 1166, 4290, 1166, 1, 1, 9357, 90002, 90002, 9357, 1, 1, 74892, 1621383, 3960088, 1621383, 74892, 1, 1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 1, 1, 4793482, 431017552, 3923731798, 7780238494, 3923731798, 431017552, 4793482, 1
Offset: 1

Views

Author

Roger L. Bagula, Sep 19 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     16,        1;
  1,    143,      143,         1;
  1,   1166,     4290,      1166,         1;
  1,   9357,    90002,     90002,      9357,        1;
  1,  74892,  1621383,   3960088,   1621383,    74892,      1;
  1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, ...
Cf. A084947 (row sums).

Programs

  • Magma
    function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A142462:= func< n,k | T(n,k,7) >;
[A142462(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022
  • Mathematica
    T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n,  1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m]];
A142462[n_, k_]:= T[n,k,7];
Table[A142462[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A142462(n,k): return T(n,k,7)
flatten([[ A142462(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 7.
Sum_{k=1..n} T(n, k) = A084947(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013

A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 179, 179, 1, 1, 1636, 6086, 1636, 1, 1, 14757, 144362, 144362, 14757, 1, 1, 132854, 2941135, 7218100, 2941135, 132854, 1, 1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1, 1, 10761672, 1001178268, 9211047544, 18315657030, 9211047544, 1001178268, 10761672, 1
Offset: 1

Views

Author

Roger L. Bagula, Nov 14 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      18,        1;
  1,     179,      179,         1;
  1,    1636,     6086,      1636,         1;
  1,   14757,   144362,    144362,     14757,        1;
  1,  132854,  2941135,   7218100,   2941135,   132854,       1;
  1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A167884, ...
Cf. A084948 (row sums).

Programs

  • Mathematica
    T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A167884[n_, k_]:= T[n,k,8];
Table[A167884[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A167884(n,k): return T(n,k,8)
flatten([[ A167884(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 8.
Sum_{k=1..n} T(n, k) = A084948(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013

A225372 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -2.

Original entry on oeis.org

1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, -4, 6, -4, 1, 1, -3, 2, 2, -3, 1, 1, -6, 15, -20, 15, -6, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1
Offset: 1

Views

Author

N. J. A. Sloane and Roger L. Bagula, May 08 2013

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1, -2,  1;
  1, -1, -1,   1;
  1, -4,  6,  -4,  1;
  1, -3,  2,   2, -3,   1;
  1, -6, 15, -20, 15,  -6,   1;
  1, -5,  9,  -5, -5,   9,  -5,  1;
  1, -8, 28, -56, 70, -56,  28, -8,  1;
  1, -7, 20, -28, 14,  14, -28, 20, -7, 1;

Links

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, A142561, A142562, A167884, ...
Cf. A130706 (row sums).

Programs

  • Magma
    function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A225372:= func< n,k | T(n,k,-2) >;
[A225372(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022
  • Maple
    T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 14 do lprint([seq(T(n,k,-2),k=1..n)]); od;
  • Mathematica
    T[n_, k_, l_] := T[n, k, l] = If[n == 1 || k == 1 || k == n, 1, (l*n-l*k+1)*T[n-1, k-1, l]+(l*k-l+1)*T[n-1, k, l]]; Table[T[n, k, -2], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)
  • Sage
    @CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A225372(n,k): return T(n,k,-2)
flatten([[ A225372(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = -2.
Sum_{k=1..n} T(n, k) = A130706(n-1). - G. C. Greubel, Mar 17 2022

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 219, 219, 1, 1, 2218, 8322, 2218, 1, 1, 22217, 220222, 220222, 22217, 1, 1, 222216, 5006247, 12332432, 5006247, 222216, 1, 1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1, 1, 22222214, 2123693776, 19700767514, 39259903390, 19700767514, 2123693776, 22222214, 1
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      20,         1;
  1,     219,       219,         1;
  1,    2218,      8322,      2218,         1;
  1,   22217,    220222,    220222,     22217,         1;
  1,  222216,   5006247,  12332432,   5006247,    222216,       1;
  1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1;

Links

Crossrefs

Cf. A084949 (row sums), A257619.
Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884
Similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,9,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257608
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,9,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

T(n, k) = t(n-k, k), where t(n,k) = f(k)*t(n-1, k) + f(n)*t(n, k-1), and f(n) = 9*n + 1.
Sum_{k=0..n} T(n, k) = A084949(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = T(n, n) = 1, a = 9, and b = 1. - G. C. Greubel, Mar 20 2022

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
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