A157278 -id:A157278 - cp's OEIS frontend

A157273 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 2, read by rows.

1, 1, 1, 1, 12, 1, 1, 47, 47, 1, 1, 154, 590, 154, 1, 1, 477, 4498, 4498, 477, 1, 1, 1448, 28323, 71232, 28323, 1448, 1, 1, 4363, 162313, 816503, 816503, 162313, 4363, 1, 1, 13110, 882764, 7897486, 15979230, 7897486, 882764, 13110, 1, 1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,     1;
  1,    12,       1;
  1,    47,      47,        1;
  1,   154,     590,      154,         1;
  1,   477,    4498,     4498,       477,         1;
  1,  1448,   28323,    71232,     28323,      1448,        1;
  1,  4363,  162313,   816503,    816503,    162313,     4363,       1;
  1, 13110,  882764,  7897486,  15979230,   7897486,   882764,   13110,     1;
  1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), A157272 (m=1), this sequence (m=2), A157274 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.

f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)

def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 2.

T(n, n-k, m) = T(n, k, m).

Edited by G. C. Greubel, Feb 05 2022

A157268 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 40, 126, 40, 1, 1, 87, 606, 606, 87, 1, 1, 182, 2413, 5856, 2413, 182, 1, 1, 373, 8679, 40337, 40337, 8679, 373, 1, 1, 756, 29376, 232726, 497066, 232726, 29376, 756, 1, 1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   17,    17,       1;
  1,   40,   126,      40,       1;
  1,   87,   606,     606,      87,       1;
  1,  182,  2413,    5856,    2413,     182,       1;
  1,  373,  8679,   40337,   40337,    8679,     373,     1;
  1,  756, 29376,  232726,  497066,  232726,   29376,   756,    1;
  1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157272, A157273, A157274, A157275, A157277, A157278.
Cf. A101945.

f[n_,k_]:= If[k<=Floor[n/2], 2^k, 2^(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 04 2022 *)

def f(n,k): return 2^k if (k <= n//2) else 2^(n-k)
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 04 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A101945(n-1), n >= 1. - G. C. Greubel, Feb 04 2022

Edited by G. C. Greubel, Feb 04 2022

A157272 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 20, 20, 1, 1, 47, 155, 47, 1, 1, 102, 753, 753, 102, 1, 1, 213, 3004, 7109, 3004, 213, 1, 1, 436, 10800, 48727, 48727, 10800, 436, 1, 1, 883, 36517, 280736, 551251, 280736, 36517, 883, 1, 1, 1778, 118795, 1454163, 4879214, 4879214, 1454163, 118795, 1778, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,    1;
  1,    7,      1;
  1,   20,     20,       1;
  1,   47,    155,      47,       1;
  1,  102,    753,     753,     102,       1;
  1,  213,   3004,    7109,    3004,     213,       1;
  1,  436,  10800,   48727,   48727,   10800,     436,      1;
  1,  883,  36517,  280736,  551251,  280736,   36517,    883,    1;
  1, 1778, 118795, 1454163, 4879214, 4879214, 1454163, 118795, 1778, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1), A157273 (m=2), A157274 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.

f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 04 2022 *)

def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 04 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 1.

T(n, n-k, m) = T(n, k, m).

Edited by G. C. Greubel, Feb 04 2022

A157274 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 84, 84, 1, 1, 355, 1431, 355, 1, 1, 1442, 14827, 14827, 1442, 1, 1, 5793, 127860, 326591, 127860, 5793, 1, 1, 23200, 1009338, 5239457, 5239457, 1009338, 23200, 1, 1, 92831, 7593061, 71229038, 145043839, 71229038, 7593061, 92831, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,     1;
  1,    17,       1;
  1,    84,      84,        1;
  1,   355,    1431,      355,         1;
  1,  1442,   14827,    14827,      1442,        1;
  1,  5793,  127860,   326591,    127860,     5793,       1;
  1, 23200, 1009338,  5239457,   5239457,  1009338,   23200,     1;
  1, 92831, 7593061, 71229038, 145043839, 71229038, 7593061, 92831, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), A157272 (m=1), A157273 (m=2), this sequence (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.

f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)

def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 3.

T(n, n-k, m) = T(n, k, m).

Edited by G. C. Greubel, Feb 05 2022

A157275 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 40, 126, 40, 1, 1, 87, 606, 606, 87, 1, 1, 182, 2413, 5604, 2413, 182, 1, 1, 373, 8679, 38117, 38117, 8679, 373, 1, 1, 756, 29376, 219020, 426002, 219020, 29376, 756, 1, 1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   17,    17,       1;
  1,   40,   126,      40,       1;
  1,   87,   606,     606,      87,       1;
  1,  182,  2413,    5604,    2413,     182,       1;
  1,  373,  8679,   38117,   38117,    8679,     373,     1;
  1,  756, 29376,  219020,  426002,  219020,   29376,   756,    1;
  1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1), A157277 (m=2), A157278 (m=3).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.
Cf. A101945.

f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)

def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m):  # A157275
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A101945(n-1), for n >= 1. - G. C. Greubel, Feb 05 2022

Edited by G. C. Greubel, Feb 05 2022

A157277 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 39, 39, 1, 1, 128, 470, 128, 1, 1, 397, 3558, 3558, 397, 1, 1, 1206, 22387, 55452, 22387, 1206, 1, 1, 3635, 128377, 632343, 632343, 128377, 3635, 1, 1, 10924, 698788, 6107192, 12269406, 6107192, 698788, 10924, 1, 1, 32793, 3686880, 53375112, 187721254, 187721254, 53375112, 3686880, 32793, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,     1;
  1,    10,      1;
  1,    39,     39,       1;
  1,   128,    470,     128,        1;
  1,   397,   3558,    3558,      397,       1;
  1,  1206,  22387,   55452,    22387,    1206,      1;
  1,  3635, 128377,  632343,   632343,  128377,   3635,     1;
  1, 10924, 698788, 6107192, 12269406, 6107192, 698788, 10924, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318 (m=0), A157275 (m=1), this sequence (m=2), A157278 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.

f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)

def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m):  # A157277
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2.

T(n, n-k, m) = T(n, k, m).

Edited by G. C. Greubel, Feb 05 2022

A157633 Triangle T(n,m) read rows: 1 in column m=0 and on the diagonal, 2m(n-m)(m^2-nm+2*n^2) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 64, 64, 1, 1, 174, 224, 174, 1, 1, 368, 528, 528, 368, 1, 1, 670, 1024, 1134, 1024, 670, 1, 1, 1104, 1760, 2064, 2064, 1760, 1104, 1, 1, 1694, 2784, 3390, 3584, 3390, 2784, 1694, 1, 1, 2464, 4144, 5184, 5680, 5680, 5184, 4144, 2464, 1, 1

Offset: 0

Roger L. Bagula, Mar 03 2009

Row sums are (30+9*n^5-10*n^3+n)/15, n>0.

			{1},
{1, 1},
{1, 14, 1},
{1, 64, 64, 1},
{1, 174, 224, 174, 1},
{1, 368, 528, 528, 368, 1},
{1, 670, 1024, 1134, 1024, 670, 1},
{1, 1104, 1760, 2064, 2064, 1760, 1104, 1},
{1, 1694, 2784, 3390, 3584, 3390, 2784, 1694, 1},
{1, 2464, 4144, 5184, 5680, 5680, 5184, 4144, 2464, 1},
{1, 3438, 5888, 7518, 8448, 8750, 8448, 7518, 5888, 3438, 1}

Cf. A157278

t[n_, m_] = If[n*m*(n - m) == 0, 1, n^4 - (m^4 + (n - m)^4)];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

T(n,m)= 1 if m=0 or m=n, else n^4 - m^4 - (n - m)^4.

Edited by the Associate Editors of the OEIS, Apr 22 2009

cp's OEIS Frontend

A157273 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 2, read by rows.

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A157268 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

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A157272 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 1, read by rows.

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A157274 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 3, read by rows.

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A157275 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 1, read by rows.

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A157277 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2, read by rows.

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A157633 Triangle T(n,m) read rows: 1 in column m=0 and on the diagonal, 2*m*(n-m)*(m^2-n*m+2*n^2) otherwise.

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A157273 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 2, read by rows.

A157268 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

A157272 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 1, read by rows.

A157274 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k+1 if k <= floor(n/2) otherwise 2(n-k)+1, and m = 3, read by rows.

A157275 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 1, read by rows.

A157277 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 2, read by rows.

A157633 Triangle T(n,m) read rows: 1 in column m=0 and on the diagonal, 2m(n-m)(m^2-nm+2*n^2) otherwise.