A157153 -id:A157153 - 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

A157147 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 110, 37, 1, 1, 83, 568, 568, 83, 1, 1, 177, 2415, 5534, 2415, 177, 1, 1, 367, 9137, 41027, 41027, 9137, 367, 1, 1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1, 1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			  1;
  1,    1;
  1,    5,      1;
  1,   15,     15,       1;
  1,   37,    110,      37,       1;
  1,   83,    568,     568,      83,       1;
  1,  177,   2415,    5534,    2415,     177,       1;
  1,  367,   9137,   41027,   41027,    9137,     367,      1;
  1,  749,  32104,  255155,  498814,  255155,   32104,    749,    1;
  1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1;

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

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), A157150 (m=4), A157151 (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

A157147:= proc(n,k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (n-k+1)*procname(n-1,k-1) +(k+1)*procname(n-1,k) +k*(n-k)*procname(n-2,k-1);
    end if;
end proc:
seq(seq(A157147(n,k),k=0..n),n=0..10); # R. J. Mathar, Feb 06 2015

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*k*(n-k)*T[n-2, k-1, m]];
Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

def T(n,k,m): # A157147
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1.

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

Edited by G. C. Greubel, Jan 09 2022

A157148 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 33, 33, 1, 1, 112, 394, 112, 1, 1, 353, 3150, 3150, 353, 1, 1, 1080, 20719, 51192, 20719, 1080, 1, 1, 3265, 122535, 620415, 620415, 122535, 3265, 1, 1, 9824, 681040, 6312360, 12805614, 6312360, 681040, 9824, 1, 1, 29505, 3643980, 57451300, 209503086, 209503086, 57451300, 3643980, 29505, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    33,      33,        1;
  1,   112,     394,      112,         1;
  1,   353,    3150,     3150,       353,         1;
  1,  1080,   20719,    51192,     20719,      1080,        1;
  1,  3265,  122535,   620415,    620415,    122535,     3265,       1;
  1,  9824,  681040,  6312360,  12805614,   6312360,   681040,    9824,     1;
  1, 29505, 3643980, 57451300, 209503086, 209503086, 57451300, 3643980, 29505, 1;

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

Cf. A007318 (m=0), A157147 (m=1), this sequence (m=2), A157149 (m=3), A157150 (m=4), A157151 (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

A157148 := proc(n,k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (2*(n-k)+1)*procname(n-1,k-1) + (2*k+1)*procname(n-1,k) + 2*k*(n-k)*procname(n-2,k-1);
    end if;
end proc:
seq(seq(A157148(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

@CachedFunction
def T(n,k,m):  # A157148
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 2.

T(n, n-k, 2) = T(n, k, 2).

Edited by G. C. Greubel, Jan 09 2022

A157149 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 57, 57, 1, 1, 247, 930, 247, 1, 1, 1013, 10006, 10006, 1013, 1, 1, 4083, 89139, 225230, 89139, 4083, 1, 1, 16369, 719691, 3771323, 3771323, 719691, 16369, 1, 1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,     1;
  1,    11,       1;
  1,    57,      57,        1;
  1,   247,     930,      247,         1;
  1,  1013,   10006,    10006,      1013,        1;
  1,  4083,   89139,   225230,     89139,     4083,       1;
  1, 16369,  719691,  3771323,   3771323,   719691,   16369,     1;
  1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 1;

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

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), this sequence (m=3), A157150 (m=4), A157151 (m=5).
Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
Cf. A289255.

A157149 := proc(n,k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (3*(n-k)+1)*procname(n-1,k-1) + (3*k+1)*procname(n-1,k) + 3*k*(n-k)*procname(n-2,k-1);
    end if;
end proc:
seq(seq(A157149(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

@CachedFunction
def T(n,k,m):  # A157149
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3.
T(n, n-k, 3) = T(n, k, 3).
T(n, 1, 3) = A289255(n). - G. C. Greubel, Jan 09 2022

Edited by G. C. Greubel, Jan 09 2022

A157150 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 87, 87, 1, 1, 460, 1790, 460, 1, 1, 2333, 24178, 24178, 2333, 1, 1, 11706, 271983, 693068, 271983, 11706, 1, 1, 58579, 2786993, 14794139, 14794139, 2786993, 58579, 1, 1, 292952, 27109300, 267169640, 547357078, 267169640, 27109300, 292952, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,      1;
  1,     14,        1;
  1,     87,       87,         1;
  1,    460,     1790,       460,         1;
  1,   2333,    24178,     24178,      2333,         1;
  1,  11706,   271983,    693068,    271983,     11706,        1;
  1,  58579,  2786993,  14794139,  14794139,   2786993,    58579,      1;
  1, 292952, 27109300, 267169640, 547357078, 267169640, 27109300, 292952, 1;

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

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), this sequence (m=4), A157151 (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

A157150:= proc(n, k);
    if k<0 or nA157150(n, k), k=0..n), n=0..10); # R. J. Mathar, Feb 06 2015

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

@CachedFunction
def T(n,k,m):  # A157150
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4.

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

Edited by G. C. Greubel, Jan 09 2022

A157151 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 123, 123, 1, 1, 769, 3046, 769, 1, 1, 4655, 49500, 49500, 4655, 1, 1, 27981, 673015, 1721070, 673015, 27981, 1, 1, 167947, 8363421, 44640435, 44640435, 8363421, 167947, 1, 1, 1007753, 98882848, 982172031, 2012583870, 982172031, 98882848, 1007753, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,       1;
  1,      17,        1;
  1,     123,      123,         1;
  1,     769,     3046,       769,          1;
  1,    4655,    49500,     49500,       4655,         1;
  1,   27981,   673015,   1721070,     673015,     27981,        1;
  1,  167947,  8363421,  44640435,   44640435,   8363421,   167947,       1;
  1, 1007753, 98882848, 982172031, 2012583870, 982172031, 98882848, 1007753, 1;

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

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), A157150 (m=4), this sequence (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

A157151:= proc(n, k)
    if k<0 or nA157151(n, k), k=0..n), n=0..10); # R. J. Mathar, Feb 06 2015

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*k*(n-k)*T[n-2, k-1, m]];
Table[T[n,k,5], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

def T(n,k,m): # A157147
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,5) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5.

T(n, n-k, 5) = T(n, k, 5).

Edited by G. C. Greubel, Jan 09 2022

A157152 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 30, 15, 1, 1, 31, 108, 108, 31, 1, 1, 63, 359, 594, 359, 63, 1, 1, 127, 1145, 2875, 2875, 1145, 127, 1, 1, 255, 3568, 12985, 19246, 12985, 3568, 255, 1, 1, 511, 10966, 56306, 116640, 116640, 56306, 10966, 511, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,    1;
  1,    3,     1;
  1,    7,     7,      1;
  1,   15,    30,     15,      1;
  1,   31,   108,    108,     31,      1;
  1,   63,   359,    594,    359,     63,      1;
  1,  127,  1145,   2875,   2875,   1145,    127,      1;
  1,  255,  3568,  12985,  19246,  12985,   3568,    255,     1;
  1,  511, 10966,  56306, 116640, 116640,  56306,  10966,   511,    1;
  1, 1023, 33417, 238024, 665702, 918530, 665702, 238024, 33417, 1023, 1;

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

Cf. A007318 (m=0), this sequence (m=1), A157153 (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).

Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

@CachedFunction
def T(n,k,m):  # A157152
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A000225(n). - G. C. Greubel, Jan 09 2022

Edited by G. C. Greubel, Jan 09 2022

A157154 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 234, 85, 1, 1, 341, 2110, 2110, 341, 1, 1, 1365, 17163, 35882, 17163, 1365, 1, 1, 5461, 131751, 505979, 505979, 131751, 5461, 1, 1, 21845, 976876, 6395471, 11433118, 6395471, 976876, 21845, 1, 1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,     1;
  1,     5,       1;
  1,    21,      21,        1;
  1,    85,     234,       85,         1;
  1,   341,    2110,     2110,       341,         1;
  1,  1365,   17163,    35882,     17163,      1365,        1;
  1,  5461,  131751,   505979,    505979,    131751,     5461,       1;
  1, 21845,  976876,  6395471,  11433118,   6395471,   976876,   21845,     1;
  1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1;

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

Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), this sequence (m=3), A157155 (m=4), A157156 (m=5).
Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
Cf. A002450.

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

@CachedFunction
def T(n,k,m):  # A157154
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 3) = A002450(n). - G. C. Greubel, Jan 10 2022

Edited by G. C. Greubel, Jan 10 2022

A157155 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 31, 31, 1, 1, 156, 462, 156, 1, 1, 781, 5442, 5442, 781, 1, 1, 3906, 57263, 124860, 57263, 3906, 1, 1, 19531, 566153, 2335435, 2335435, 566153, 19531, 1, 1, 97656, 5396164, 38814088, 71413750, 38814088, 5396164, 97656, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,     1;
  1,     6,       1;
  1,    31,      31,        1;
  1,   156,     462,      156,        1;
  1,   781,    5442,     5442,      781,        1;
  1,  3906,   57263,   124860,    57263,     3906,       1;
  1, 19531,  566153,  2335435,  2335435,   566153,   19531,     1;
  1, 97656, 5396164, 38814088, 71413750, 38814088, 5396164, 97656, 1;

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

Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), A157154 (m=3), this sequence (m=4), A157156 (m=5).
Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
Cf. A003463.

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

@CachedFunction
def T(n,k,m):  # A157155
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 4) = A003463(n). - G. C. Greubel, Jan 10 2022

Edited by G. C. Greubel, Jan 10 2022

A157156 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 43, 1, 1, 259, 806, 259, 1, 1, 1555, 11720, 11720, 1555, 1, 1, 9331, 151215, 338770, 151215, 9331, 1, 1, 55987, 1828221, 7892635, 7892635, 1828221, 55987, 1, 1, 335923, 21286168, 162474781, 304389070, 162474781, 21286168, 335923, 1

Offset: 0

Roger L. Bagula, Feb 24 2009

			Triangle begins as:
  1;
  1,      1;
  1,      7,        1;
  1,     43,       43,         1;
  1,    259,      806,       259,         1;
  1,   1555,    11720,     11720,      1555,         1;
  1,   9331,   151215,    338770,    151215,      9331,        1;
  1,  55987,  1828221,   7892635,   7892635,   1828221,    55987,      1;
  1, 335923, 21286168, 162474781, 304389070, 162474781, 21286168, 335923, 1;

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

Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).
Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
Cf. A003464.

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*k*(n-k)*T[n-2,k-1,m]];
Table[T[n,k,5], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

@CachedFunction
def T(n,k,m):  # A157156
    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*k*(n-k)*T(n-2,k-1,m)
flatten([[T(n,k,5) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 5) = A003464(n). - G. C. Greubel, Jan 10 2022

Edited by G. C. Greubel, Jan 10 2022

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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A157147 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

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A157148 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 2, read by rows.

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A157149 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

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A157150 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

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A157151 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.

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A157152 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

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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.

A157147 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

A157148 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 2, read by rows.

A157149 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

A157150 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

A157151 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.

A157152 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.

A157154 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

A157155 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

A157156 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) - mk(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.