A157207 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 = 1, read by rows.

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 33, 94, 33, 1, 1, 72, 442, 442, 72, 1, 1, 151, 1752, 3818, 1752, 151, 1, 1, 310, 6306, 25358, 25358, 6306, 310, 1, 1, 629, 21390, 144524, 268852, 144524, 21390, 629, 1, 1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 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) = k if k <= floor(n/2) otherwise n-k, and m = 1.

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

T(n, 1, 1) = A094002(n-1). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   14,    14,      1;
  1,   33,    94,     33,       1;
  1,   72,   442,    442,      72,       1;
  1,  151,  1752,   3818,    1752,     151,      1;
  1,  310,  6306,  25358,   25358,    6306,    310,     1;
  1,  629, 21390, 144524,  268852,  144524,  21390,   629,    1;
  1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 1;

Mathematica

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, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

Prog

(Sage)
def f(n, k): return k if (k <= n//2) else 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..20)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A157208 (m=2), A157209 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Cf. A094002.

Sequence in context: A111910 A181143 A144438 * A008957 A136267 A109960

Adjacent sequences: A157204 A157205 A157206 * A157208 A157209 A157210

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved