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

0,5

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 = 2.

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

Example

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;

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 *)

Prog

(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

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.

Sequence in context: A178347 A141686 A185412 * A220595 A154335 A142467

Adjacent sequences: A157145 A157146 A157147 * A157149 A157150 A157151

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved