A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 33, 89, 33, 1, 1, 72, 413, 413, 72, 1, 1, 151, 1632, 3393, 1632, 151, 1, 1, 310, 5874, 22145, 22145, 5874, 310, 1, 1, 629, 19943, 125456, 224843, 125456, 19943, 629, 1, 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n,k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1), T(n, 1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = A001053(n+1).

From G. C. Greubel, Mar 03 2022: (Start)

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

T(n, 3) = (1/2)*(n^2 +3*n +1 + 73*3^(n-3) - 5*2^(n-2)*(2*n+3)). (End)

Example

Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   14,    14,      1;
  1,   33,    89,     33,       1;
  1,   72,   413,    413,      72,       1;
  1,  151,  1632,   3393,    1632,     151,      1;
  1,  310,  5874,  22145,   22145,    5874,    310,     1;
  1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
  1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;

Mathematica

T[n_, k_, m_, j_]:= T[n, k, m, j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m, j] + (m*(k-1)+1)*T[n-1, k, m, j] + j*T[n-2, k-1, m, j]];
Table[T[n, k, 1, 1], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)

Prog

(Sage)
def T(n, k, m, j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m, j) + (m*(k-1)+1)*T(n-1, k, m, j) + j*T(n-2, k-1, m, j)
def A144438(n, k): return T(n, k, 1, 1)
flatten([[A144438(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Crossrefs

Cf. A001053 (row sums), A094002 (column k=2).

Cf. A144431, A144432, A144435, A144436.

Sequence in context: A278880 A111910 A181143 * A157207 A008957 A136267

Adjacent sequences: A144435 A144436 A144437 * A144439 A144440 A144441

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Status

approved