A157151 -id:A157151 - cp's OEIS frontend

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.

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

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

cp's OEIS Frontend

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.

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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.

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Author

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Programs

Mathematica

Sage

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Extensions

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

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.

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.