A157152 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, 3, 1, 1, 7, 7, 1, 1, 15, 30, 15, 1, 1, 31, 108, 108, 31, 1, 1, 63, 359, 594, 359, 63, 1, 1, 127, 1145, 2875, 2875, 1145, 127, 1, 1, 255, 3568, 12985, 19246, 12985, 3568, 255, 1, 1, 511, 10966, 56306, 116640, 116640, 56306, 10966, 511, 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 = 5.

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

T(n, 1, 1) = A000225(n). - G. C. Greubel, Jan 09 2022

Example

Triangle begins as:
  1;
  1,    1;
  1,    3,     1;
  1,    7,     7,      1;
  1,   15,    30,     15,      1;
  1,   31,   108,    108,     31,      1;
  1,   63,   359,    594,    359,     63,      1;
  1,  127,  1145,   2875,   2875,   1145,    127,      1;
  1,  255,  3568,  12985,  19246,  12985,   3568,    255,     1;
  1,  511, 10966,  56306, 116640, 116640,  56306,  10966,   511,    1;
  1, 1023, 33417, 238024, 665702, 918530, 665702, 238024, 33417, 1023, 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, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157152
    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..20)]) # G. C. Greubel, Jan 09 2022

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A157153 (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.

Sequence in context: A193871 A108470 A328300 * A136126 A046802 A184173

Adjacent sequences: A157149 A157150 A157151 * A157153 A157154 A157155

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved