A157153 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, 4, 1, 1, 13, 13, 1, 1, 40, 98, 40, 1, 1, 121, 614, 614, 121, 1, 1, 364, 3519, 6832, 3519, 364, 1, 1, 1093, 19179, 64759, 64759, 19179, 1093, 1, 1, 3280, 101368, 558712, 947038, 558712, 101368, 3280, 1, 1, 9841, 525436, 4538324, 12078814, 12078814, 4538324, 525436, 9841, 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, m) = T(n, k, m) for m = 2.

T(n, 1, 2) = A003462(n). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,       1;
  1,   40,     98,      40,        1;
  1,  121,    614,     614,      121,        1;
  1,  364,   3519,    6832,     3519,      364,       1;
  1, 1093,  19179,   64759,    64759,    19179,    1093,      1;
  1, 3280, 101368,  558712,   947038,   558712,  101368,   3280,    1;
  1, 9841, 525436, 4538324, 12078814, 12078814, 4538324, 525436, 9841, 1;

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 10 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157153
    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 10 2022

Crossrefs

Cf. A007318 (m=0), A157152 (m=1), this sequence (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).

Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Cf. A003462.

Sequence in context: A255494 A146956 A152613 * A212801 A147565 A022167

Adjacent sequences: A157150 A157151 A157152 * A157154 A157155 A157156

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved