A157148 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 = 2, read by rows.

1, 1, 1, 1, 8, 1, 1, 33, 33, 1, 1, 112, 394, 112, 1, 1, 353, 3150, 3150, 353, 1, 1, 1080, 20719, 51192, 20719, 1080, 1, 1, 3265, 122535, 620415, 620415, 122535, 3265, 1, 1, 9824, 681040, 6312360, 12805614, 6312360, 681040, 9824, 1, 1, 29505, 3643980, 57451300, 209503086, 209503086, 57451300, 3643980, 29505, 1

Offset: 0

Author

Roger L. Bagula, Feb 24 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    33,      33,        1;
  1,   112,     394,      112,         1;
  1,   353,    3150,     3150,       353,         1;
  1,  1080,   20719,    51192,     20719,      1080,        1;
  1,  3265,  122535,   620415,    620415,    122535,     3265,       1;
  1,  9824,  681040,  6312360,  12805614,   6312360,   681040,    9824,     1;
  1, 29505, 3643980, 57451300, 209503086, 209503086, 57451300, 3643980, 29505, 1;
		

Links

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

Crossrefs

Cf. A007318 (m=0), A157147 (m=1), this sequence (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

A157148 := proc(n,k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (2*(n-k)+1)*procname(n-1,k-1) + (2*k+1)*procname(n-1,k) + 2*k*(n-k)*procname(n-2,k-1);
    end if;
end proc:
seq(seq(A157148(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,2], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

Sage

@CachedFunction
def T(n,k,m):  # A157148
    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,2) for k in (0..n)] for n in (0..20)]) # 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 = 2.

T(n, n-k, 2) = T(n, k, 2).

Extensions

Edited by G. C. Greubel, Jan 09 2022