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

0,5

Links

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

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).

Example

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;

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 *)

Prog

(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

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.

Sequence in context: A176491 A008958 A168524 * A157629 A154336 A174109

Adjacent sequences: A157274 A157275 A157276 * A157278 A157279 A157280

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 05 2022

Status

approved