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.

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

Author

Roger L. Bagula, Feb 26 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

Sage

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

Formula

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).

Extensions

Edited by G. C. Greubel, Feb 05 2022