A157277 -id:A157277 - 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

A157278 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 = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 69, 69, 1, 1, 292, 1134, 292, 1, 1, 1187, 11686, 11686, 1187, 1, 1, 4770, 100737, 254132, 100737, 4770, 1, 1, 19105, 795723, 4061249, 4061249, 795723, 19105, 1, 1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,     1;
  1,    14,       1;
  1,    69,      69,        1;
  1,   292,    1134,      292,         1;
  1,  1187,   11686,    11686,      1187,        1;
  1,  4770,  100737,   254132,    100737,     4770,       1;
  1, 19105,  795723,  4061249,   4061249,   795723,   19105,     1;
  1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1;

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

Cf. A007318 (m=0), A157275 (m=1), A157277 (m=2), this sequence (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,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 06 2022 *)

def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m):  # A157278
    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 06 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 = 3.

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

Edited by G. C. Greubel, Feb 06 2022

A157629 A general recursion triangle with third part a power triangle:m=2; Power triangle: f(n,k,m)=If[nk(n - k) == 0, 1, n^m - (k^m + (n - k)^m)]; Recursion: A(n,k,m)=(m(n - k) + 1)A(n - 1, k - 1, m) + (mk + 1)A(n - 1, k, m) + mf(n, k, m)A(n - 2, k - 1, m).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 43, 43, 1, 1, 148, 590, 148, 1, 1, 469, 5018, 5018, 469, 1, 1, 1438, 34047, 91492, 34047, 1438, 1, 1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1, 1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1, 1, 39337

Offset: 0

Roger L. Bagula, Mar 03 2009

Row sums are:

{1, 2, 12, 88, 888, 10976, 162464, 2793792, 54779904, 1206055680, 29460493056,...}.

			{1},
{1, 1},
{1, 10, 1},
{1, 43, 43, 1},
{1, 148, 590, 148, 1},
{1, 469, 5018, 5018, 469, 1},
{1, 1438, 34047, 91492, 34047, 1438, 1},
{1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1},
{1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1},
{1, 39337, 6188356, 117961172, 478838974, 478838974, 117961172, 6188356, 39337, 1},
{1, 118066, 32448653, 1015124312, 7053594482, 13257922028, 7053594482, 1015124312, 32448653, 118066, 1}

A142459, A154336, A157277

A[n_, 0, m_] := 1; A[n_, n_, m_] := 1;
A[n_, k_, m_] := (m*(n - k) + 1)*A[n - 1, k - 1, m] + (m*k + 1)*A[n - 1, k, m] + m*f[n, k, m]*A[n - 2, k - 1, m];
Table[A[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[A[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[A[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

m=0:Pascal:m=1Eulerian numbers;
m=2;
Power triangle:
f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)];
Recursion:
A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) +
(m*k + 1)*A(n - 1, k, m) +
m*f(n, k, m)*A(n - 2, k - 1, m).

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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A157278 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 = 3, read by rows.

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A157629 A general recursion triangle with third part a power triangle:m=2; Power triangle: f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)]; Recursion: A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) + (m*k + 1)*A(n - 1, k, m) + m*f(n, k, m)*A(n - 2, k - 1, m).

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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.

A157278 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 = 3, read by rows.

A157629 A general recursion triangle with third part a power triangle:m=2; Power triangle: f(n,k,m)=If[nk(n - k) == 0, 1, n^m - (k^m + (n - k)^m)]; Recursion: A(n,k,m)=(m(n - k) + 1)A(n - 1, k - 1, m) + (mk + 1)A(n - 1, k, m) + mf(n, k, m)A(n - 2, k - 1, m).