A132308 -id:A132308 - cp's OEIS frontend

A157211 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.

1, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 50, 134, 50, 1, 1, 157, 960, 960, 157, 1, 1, 480, 6013, 12636, 6013, 480, 1, 1, 1451, 34717, 136809, 136809, 34717, 1451, 1, 1, 4366, 190528, 1303472, 2361474, 1303472, 190528, 4366, 1, 1, 13113, 1012326, 11392866, 34496986, 34496986, 11392866, 1012326, 13113, 1

Offset: 0

Roger L. Bagula, Feb 25 2009

			Triangle begins as:
  1;
  1,     1;
  1,     4,       1;
  1,    15,      15,        1;
  1,    50,     134,       50,        1;
  1,   157,     960,      960,      157,        1;
  1,   480,    6013,    12636,     6013,      480,        1;
  1,  1451,   34717,   136809,   136809,    34717,     1451,       1;
  1,  4366,  190528,  1303472,  2361474,  1303472,   190528,    4366,     1;
  1, 13113, 1012326, 11392866, 34496986, 34496986, 11392866, 1012326, 13113, 1;

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

Cf. A007318 (m=0), A157210 (m=1), this sequence (m=2), A157212 (m=3).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.
Cf. A132308.

f[n_,k_]:= If[k<=Floor[n/2], k, 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, Jan 10 2022 *)

def f(n,k): return k if (k <= n//2) else n-k
@CachedFunction
def T(n,k,m):  # A157211
    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, 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*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 2.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 2) = A132308(n-1). - G. C. Greubel, Jan 10 2022

Edited by G. C. Greubel, Jan 10 2022

A132307 2*A007318^(2) - A000012.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 23, 11, 1, 31, 63, 47, 15, 1, 63, 159, 159, 79, 19, 1, 127, 383, 479, 319, 119, 23, 1, 255, 895, 1343, 1119, 559, 167, 27, 1, 511, 2047, 3583, 3583, 2239, 895, 223, 31, 1, 1023, 4607, 9215, 10751, 8063, 4031, 1343, 287, 35, 1

Offset: 0

Gary W. Adamson, Aug 18 2007

Row sums = A132308: (1, 4, 15, 50, 157, 480, 1451, ...). Inverse binomial transform of A132307 = triangle A132309 (having row sums A077552).

			First few rows of the triangle:
   1;
   3,   1;
   7,   7,   1;
  15,  23,  11,  1;
  31,  63,  47, 15,  1;
  63, 159, 159, 79, 19, 1;
  ...

Cf. A132308, A132309, A077552.

2*A007318^(2) - A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A157211 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.

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A132307 2*A007318^(2) - A000012.

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cp's OEIS Frontend

A157211 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.

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Author

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Programs

Mathematica

Sage

Formula

Extensions

A132307 2*A007318^(2) - A000012.

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Comments

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Crossrefs

Formula

A157211 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.