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

Author

Roger L. Bagula, Feb 24 2009

Keywords

• nonn
• tabl
• easy

Examples

			  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;
		

Links

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

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.

Programs

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 *)

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

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).

Extensions

Edited by G. C. Greubel, Jan 09 2022