A157147 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 = 1, read by rows.

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 110, 37, 1, 1, 83, 568, 568, 83, 1, 1, 177, 2415, 5534, 2415, 177, 1, 1, 367, 9137, 41027, 41027, 9137, 367, 1, 1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1, 1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 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 = 1.

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

Example

  1;
  1,    1;
  1,    5,      1;
  1,   15,     15,       1;
  1,   37,    110,      37,       1;
  1,   83,    568,     568,      83,       1;
  1,  177,   2415,    5534,    2415,     177,       1;
  1,  367,   9137,   41027,   41027,    9137,     367,      1;
  1,  749,  32104,  255155,  498814,  255155,   32104,    749,    1;
  1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1;

Maple

A157147:= proc(n, k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (n-k+1)*procname(n-1, k-1) +(k+1)*procname(n-1, k) +k*(n-k)*procname(n-2, k-1);
    end if;
end proc:
seq(seq(A157147(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, 1], {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, 1) 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), A157151 (m=5).

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

Sequence in context: A168288 A157523 A141691 * A347973 A232103 A292357

Adjacent sequences: A157144 A157145 A157146 * A157148 A157149 A157150

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved