A157151 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5, read by rows.

1, 1, 1, 1, 17, 1, 1, 123, 123, 1, 1, 769, 3046, 769, 1, 1, 4655, 49500, 49500, 4655, 1, 1, 27981, 673015, 1721070, 673015, 27981, 1, 1, 167947, 8363421, 44640435, 44640435, 8363421, 167947, 1, 1, 1007753, 98882848, 982172031, 2012583870, 982172031, 98882848, 1007753, 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*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 5.

T(n, n-k, 5) = T(n, k, 5).

Example

Triangle begins as:
  1;
  1,       1;
  1,      17,        1;
  1,     123,      123,         1;
  1,     769,     3046,       769,          1;
  1,    4655,    49500,     49500,       4655,         1;
  1,   27981,   673015,   1721070,     673015,     27981,        1;
  1,  167947,  8363421,  44640435,   44640435,   8363421,   167947,       1;
  1, 1007753, 98882848, 982172031, 2012583870, 982172031, 98882848, 1007753, 1;

Maple

A157151:= proc(n, k)
    if k<0 or n<k then 0;
    elif k=0 or k=n then 1;
    else (5*n-5*k+1)*procname(n-1, k-1) + (5*k+1)*procname(n-1, k) + 5*k*(n-k)*procname(n-2, k-1);
    end if; end proc;
seq(seq(A157151(n, k), k=0..n), n=0..10); # R. J. Mathar, Feb 06 2015

Mathematica

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*k*(n-k)*T[n-2, k-1, m]];
Table[T[n, k, 5], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

Prog

(Sage)
def T(n, k, m): # A157147
    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*k*(n-k)*T(n-2, k-1, m)
flatten([[T(n, k, 5) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 09 2022

Crossrefs

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), A157150 (m=4), this sequence (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Sequence in context: A157274 A218115 A144442 * A176794 A176244 A022180

Adjacent sequences: A157148 A157149 A157150 * A157152 A157153 A157154

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved